Research Article A Generalization on Weighted Means and Convex Functions with respect to the Non-Newtonian Calculus ULur Kadak 1 and Yusuf Gürefe 2 1 Department of Mathematics, Bozok University, 66100 Yozgat, Turkey 2 Department of Econometrics, Us ¸ak University, 64300 Us ¸ak, Turkey Correspondence should be addressed to U˘ gur Kadak; [email protected] Received 22 July 2016; Accepted 19 September 2016 Academic Editor: Julien Salomon Copyright © 2016 U. Kadak and Y. G¨ urefe. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper is devoted to investigating some characteristic features of weighted means and convex functions in terms of the non- Newtonian calculus which is a self-contained system independent of any other system of calculus. It is shown that there are infinitely many such useful types of weighted means and convex functions depending on the choice of generating functions. Moreover, some relations between classical weighted mean and its non-Newtonian version are compared and discussed in a table. Also, some geometric interpretations of convex functions are presented with respect to the non-Newtonian slope. Finally, using multiplicative continuous convex functions we give an application. 1. Introduction It is well known that the theory of convex functions and weighted means plays a very important role in mathematics and other fields. ere is wide literature covering this topic (see, e.g., [1–8]). Nowadays the study of convex functions has evolved into a larger theory about functions which are adapted to other geometries of the domain and/or obey other laws of comparison of means. Also the study of convex functions begins in the context of real-valued functions of a real variable. More important, they will serve as a model for deep generalizations into the setting of several variables. As an alternative to the classical calculus, Grossman and Katz [9–11] introduced the non-Newtonian calculus consist- ing of the branches of geometric, quadratic and harmonic calculus, and so forth. All these calculi can be described simultaneously within the framework of a general theory. ey decided to use the adjective non-Newtonian to indicate any calculi other than the classical calculus. Every property in classical calculus has an analogue in non-Newtonian calculus which is a methodology that allows one to have a different look at problems which can be investigated via calculus. In some cases, for example, for wage-rate (in dollars, euro, etc.) related problems, the use of bigeometric calculus which is a kind of non-Newtonian calculus is advocated instead of a traditional Newtonian one. Many authors have extensively developed the notion of multiplicative calculus; see [12–14] for details. Also some authors have also worked on the classical sequence spaces and related topics by using non-Newtonian calculus [15–17]. Furthermore, Kadak et al. [18, 19] characterized the classes of matrix transformations between certain sequence spaces over the non-Newtonian complex field and generalized Runge- Kutta method with respect to the non-Newtonian calculus. For more details, see [20–22]. e main focus of this work is to extend weighted means and convex functions based on various generator functions, that is, exp and ( ∈ R + ) generators. e rest of this paper is organized as follows: in Sec- tion 2, we give some required definitions and consequences related with the -arithmetic and -arithmetic. Based on two arbitrarily selected generators and , we give some basic definitions with respect to the ∗-arithmetic. We also report the most relevant and recent literature in this section. Hindawi Publishing Corporation International Journal of Analysis Volume 2016, Article ID 5416751, 9 pages http://dx.doi.org/10.1155/2016/5416751
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Research ArticleA Generalization on Weighted Means and Convex Functionswith respect to the Non-Newtonian Calculus
ULur Kadak1 and Yusuf Guumlrefe2
1Department of Mathematics Bozok University 66100 Yozgat Turkey2Department of Econometrics Usak University 64300 Usak Turkey
Correspondence should be addressed to Ugur Kadak ugurkadakgmailcom
Received 22 July 2016 Accepted 19 September 2016
Academic Editor Julien Salomon
Copyright copy 2016 U Kadak and Y Gurefe This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper is devoted to investigating some characteristic features of weighted means and convex functions in terms of the non-Newtonian calculus which is a self-contained system independent of any other system of calculus It is shown that there are infinitelymany such useful types of weighted means and convex functions depending on the choice of generating functions Moreoversome relations between classical weighted mean and its non-Newtonian version are compared and discussed in a table Also somegeometric interpretations of convex functions are presented with respect to the non-Newtonian slope Finally using multiplicativecontinuous convex functions we give an application
1 Introduction
It is well known that the theory of convex functions andweighted means plays a very important role in mathematicsand other fields There is wide literature covering this topic(see eg [1ndash8]) Nowadays the study of convex functionshas evolved into a larger theory about functions which areadapted to other geometries of the domain andor obeyother laws of comparison of means Also the study of convexfunctions begins in the context of real-valued functions of areal variable More important they will serve as a model fordeep generalizations into the setting of several variables
As an alternative to the classical calculus Grossman andKatz [9ndash11] introduced the non-Newtonian calculus consist-ing of the branches of geometric quadratic and harmoniccalculus and so forth All these calculi can be describedsimultaneously within the framework of a general theoryThey decided to use the adjective non-Newtonian to indicateany calculi other than the classical calculus Every property inclassical calculus has an analogue in non-Newtonian calculuswhich is a methodology that allows one to have a differentlook at problems which can be investigated via calculus In
some cases for example for wage-rate (in dollars euro etc)related problems the use of bigeometric calculus which isa kind of non-Newtonian calculus is advocated instead of atraditional Newtonian one
Many authors have extensively developed the notion ofmultiplicative calculus see [12ndash14] for details Also someauthors have also worked on the classical sequence spacesand related topics by using non-Newtonian calculus [15ndash17]Furthermore Kadak et al [18 19] characterized the classes ofmatrix transformations between certain sequence spaces overthe non-Newtonian complex field and generalized Runge-Kutta method with respect to the non-Newtonian calculusFor more details see [20ndash22]
The main focus of this work is to extend weightedmeans and convex functions based on various generatorfunctions that is exp and 119902119901 (119901 isin R+) generators
The rest of this paper is organized as follows in Sec-tion 2 we give some required definitions and consequencesrelated with the 120572-arithmetic and 119902119901-arithmetic Based ontwo arbitrarily selected generators 120572 and 120573 we give somebasic definitions with respect to the lowast-arithmetic We alsoreport the most relevant and recent literature in this section
Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2016 Article ID 5416751 9 pageshttpdxdoiorg10115520165416751
2 International Journal of Analysis
In Section 3 first the definitions of non-Newtonian meansare given which will be used for non-Newtonian convexity Inthis section the forms of weighted means are presented andan illustrative table is given In Section 4 the generalized non-Newtonian convex function is defined on the interval 119868120572 andsome types of convex function are obtained by using differentgenerators In the final section of the paper we assert thenotion of multiplicative Lipschitz condition on the closedinterval [119909 119910] sub (0infin)2 Preliminary Background and Notation
Arithmetic is any system that satisfies the whole of theordered field axioms whose domain is a subset of R Thereare infinitely many types of arithmetic all of which areisomorphic that is structurally equivalent
A generator 120572 is a one-to-one function whose domain isR and whose range is a subset R120572 of R where R120572 = 120572(119909) 119909 isin R Each generator generates exactly one arithmeticand conversely each arithmetic is generated by exactly onegenerator If 119868(119909) = 119909 for all 119909 isin R the identity functionrsquosinverse is itself In the special cases 120572 = 119868 and 120572 = exp 120572generates the classical and geometric arithmetic respectivelyBy 120572-arithmetic we mean the arithmetic whose domain is Rand whose operations are defined as follows for 119909 119910 isin R120572and any generator 120572
120572-addition119909 + 119910 = 120572 120572minus1 (119909) + 120572minus1 (119910) 120572-subtraction119909 minus 119910 = 120572 120572minus1 (119909) minus 120572minus1 (119910)
120572-multiplication119909 times 119910 = 120572 120572minus1 (119909) times 120572minus1 (119910) 120572-division 119909 119910 = 120572 120572minus1 (119909) divide 120572minus1 (119910)
120572-order 119909 lt 119910 lArrrArr 120572minus1 (119909) lt 120572minus1 (119910)
(1)
As a generator we choose exp function acting from R intothe set Rexp = (0infin) as follows
120572 R 997888rarr Rexp
119909 997891997888rarr 119910 = 120572 (119909) = 119890119909 (2)
It is obvious that 120572-arithmetic reduces to the geometric arith-metic as follows
geometric addition 119909 + 119910 = 119890ln119909+ln119910 = 119909 sdot 119910geometric subtraction 119909 minus 119910 = 119890ln119909minusln119910 = 119909 divide 119910geometric multiplication 119909 times 119910 = 119890ln 119909 ln119910 = 119909ln119910= 119910ln119909
geometric division 119909 119910 = 119890ln 119909 ln119910 = 1199091 ln119910geometric order 119909 lt 119910 lArrrArr ln (119909) lt ln (119910)
(3)
Following Grossman and Katz [10] we give the infinitelymany 119902119901-arithmetics of which the quadratic and harmonic
arithmetic are special cases for119901 = 2 and119901 = minus1 respectivelyThe function 119902119901 Rrarr R119902 sube R and its inverse 119902minus1119901 are definedas follows (119901 isin R 0)
119902119901 (119909) =
1199091119901 119909 gt 00 119909 = 0minus (minus119909)1119901 119909 lt 0
119902minus1119901 (119909) =
119909119901 119909 gt 00 119909 = 0minus (minus119909)119901 119909 lt 0
(4)
It is to be noted that 119902119901-calculus is reduced to the classicalcalculus for 119901 = 1 Additionally it is concluded that the 120572-summation can be given as follows
119899
120572sum119896=1
119909119896 = 120572 119899sum119896=1
120572minus1 (119909119896)= 120572 120572minus1 (1199091) + sdot sdot sdot + 120572minus1 (119909119899) forall119909119896 isin R
+(5)
Definition 1 (see [15]) Let119883 = (119883 119889120572) be an 120572-metric spaceThen the basic notions can be defined as follows
(a) A sequence 119909 = (119909119896) is a function from the set N intothe setR120572 The 120572-real number 119909119896 denotes the value ofthe function at 119896 isin N and is called the 119896th term of thesequence
(b) A sequence (119909119899) in 119883 = (119883 119889120572) is said to be 120572-convergent if for every given 120576 gt 0 (120576 isin R120572) thereexist an 1198990 = 1198990(120576) isin N and 119909 isin 119883 such that119889120572(119909119899 119909) = |119909119899 minus 119909|120572 lt 120576 for all 119899 gt 1198990 and is denotedby 120572lim119899rarrinfin119909119899 = 119909 or 119909119899 120572997888rarr 119909 as 119899 rarr infin
(c) A sequence (119909119899) in119883 = (119883 119889120572) is said to be 120572-Cauchyif for every 120576 gt 0 there is an 1198990 = 1198990(120576) isin N such that119889120572(119909119899 119909119898) lt 120576 for all119898 119899 gt 1198990
Throughout this paper we define the 119901th 120572-exponent 119909119901120572and 119902th 120572-root 119909(1119902)120572 of 119909 isin R+ by
1199092120572 = 119909 times 119909 = 120572 120572minus1 (119909) times 120572minus1 (119909)= 120572 [120572minus1 (119909)]2
1199093120572 = 1199092120572 times 119909= 120572 120572minus1 120572 [120572minus1 (119909) times 120572minus1 (119909)] times 120572minus1 (119909)= 120572 [120572minus1 (119909)]3
119909119901120572 = 119909(119901minus1)120572 times 119909 = 120572 [120572minus1 (119909)]119901
(6)
International Journal of Analysis 3
and 120572radic119909 = 119909(12)120572 = 119910 provided there exists an 119910 isin R120572 suchthat 1199102120572 = 11990921 lowast-Arithmetic Suppose that 120572 and 120573 are two arbitrarilyselected generators and (ldquostar-rdquo) also is the ordered pairof arithmetics (120573-arithmetic and 120572-arithmetic) The sets(R120573 + minus times lt) and (R120572 + minus times lt) are complete orderedfields and 119887119890119905119886(119886119897119901ℎ119886)-generator generates 119887119890119905119886(119886119897119901ℎ119886)-arithmetic respectively Definitions given for 120573-arithmeticare also valid for 120572-arithmetic Also 120572-arithmetic is used forarguments and 120573-arithmetic is used for values in particu-lar changes in arguments and values are measured by 120572-differences and 120573-differences respectively
Let 119909 isin (R120572 + minus times lt) and 119910 isin (R120573 + minus times lt) bearbitrarily chosen elements from corresponding arithmeticThen the ordered pair (119909 119910) is called a lowast-point and the setof all lowast-points is called the set of lowast-complex numbers and isdenoted by Clowast that is
Clowast fl 119911lowast = (119909 119910) | 119909 isin R120572 119910 isin R120573 (7)
Definition 2 (see [17]) (a) The lowast-limit of a function 119891 at anelement 119886 inR120572 is if it exists the unique number 119887 inR120573 suchthatlowast lim119909rarr119886
119891 (119909) = 119887 lArrrArrforall120576 gt 0 exist120575 gt 0 ni 1003816100381610038161003816119891 (119909) minus 1198871003816100381610038161003816120573 lt 120576
forall119909 120576 isin R120572 1003816100381610038161003816119909 minus 1198861003816100381610038161003816120572 lt 120575(8)
for 120575 isin R120573 and is written as lowastlim119909rarr119886119891(119909) = 119887A function119891 islowast-continuous at a point 119886 inR120572 if and only
if 119886 is an argument of 119891 and lowastlim119909rarr119886119891(119909) = 119891(119886) When120572 and 120573 are the identity function 119868 the concepts of lowast-limitand lowast-continuity are reduced to those of classical limit andclassical continuity
(b) The isomorphism from 120572-arithmetic to 120573-arithmeticis the unique function 120580 (iota) which has the following threeproperties
(i) 120580 is one to one(ii) 120580 is from R120572 to R120573
(iii) For any numbers 119906 V isin R120572
120580 (119906 + V) = 120580 (119906) + 120580 (V) 120580 (119906 minus V) = 120580 (119906) minus 120580 (V) 120580 (119906 times V) = 120580 (119906) times 120580 (V) 120580 (119906 V) = 120580 (119906) 120580 (V)
(9)
It turns out that 120580(119909) = 120573120572minus1(119909) for every 119909 in R120572 andthat 120580() = for every 120572-integer Since for example119906 + V = 120580minus1120580(119906) + 120580(V) it should be clear that any statement in
120572-arithmetic can readily be transformed into a statement in120573-arithmetic
Definition 3 (see [10]) The following statements are valid
(i) The lowast-points 1198751 1198752 and 1198753 are lowast-collinear providedthat at least one of the following holds
119889lowast (1198752 1198751) + 119889lowast (1198751 1198753) = 119889lowast (1198752 1198753) 119889lowast (1198751 1198752) + 119889lowast (1198752 1198753) = 119889lowast (1198751 1198753) 119889lowast (1198751 1198753) + 119889lowast (1198753 1198752) = 119889lowast (1198751 1198752)
(10)
(ii) A lowast-line is a set 119871 of at least two distinct points suchthat for any distinct points1198751 and1198752 in 119871 a point1198753 isin 119871 if and only if 1198751 1198752 and 1198753 are lowast-collinear When120572 = 120573 = 119868 the lowast-lines are the straight lines in two-dimensional Euclidean space
(iii) The lowast-slope of a lowast-line through the points (1198861 1198871) and(1198862 1198872) is given by
119898lowast = (1198872 minus 1198871) 120580 (1198862 minus 1198861)= 120573 120573minus1 (1198872) minus 120573minus1 (1198871)120572minus1 (1198862) minus 120572minus1 (1198861) (1198861 = 1198862) (11)
for 1198861 1198862 isin R120572 and 1198871 1198872 isin R120573
If the following lowast-limit in (12) exists we denote it by119891lowast(119905) call it the lowast-derivative of 119891 at 119905 and say that 119891 is lowast-differentiable at 119905 (see [19])
lowastlim119909rarr119905
(119891 (119909) minus 119891 (119905)) 120580 (119909 minus 119905)= lim119909rarr119905
120573120573minus1 119891 (119909) minus 120573minus1 119891 (119905)120572minus1 (119909) minus 120572minus1 (119905)
= lim119909rarr119905
120573120573minus1 119891 (119909) minus 120573minus1 119891 (119905)119909 minus 119905sdot 119909 minus 119905120572minus1 (119909) minus 120572minus1 (119905) = 120573
(120573minus1 ∘ 119891)1015840 (119905)(120572minus1)1015840 (119905)
(12)
3 Non-Newtonian (Weighted) Means
Definition 4 (120572-arithmetic mean) Consider that 119899 positivereal numbers 1199091 1199092 119909119899 are given The 120572-mean (average)
4 International Journal of Analysis
denoted by 119860120572 is the 120572-sum of 119909119899rsquos 120572-divided by for all119899 isin N That is
119860120572 = 119899
120572sum119896=1
119909119896 = 119899
120572sum119896=1
120572120572minus1 (119909119896)119899
= 120572120572minus1 (1199091) + 120572minus1 (1199092) + sdot sdot sdot + 120572minus1 (119909119899)119899 (13)
For 120572 = exp we obtain that
119860exp = ( 119899prod119896=1
119909119896)1119899
= (1199091 sdot 1199092 sdot sdot sdot 119909119899)1119899 (14)
Similarly for 120572 = 119902119901 we get
119860119901 = (1199091199011 + 1199091199012 + sdot sdot sdot + 119909119901119899119899 )1119901 119901 isin R 0 (15)
119860exp and 119860119901 are called multiplicative arithmetic mean and119901-arithmetic mean (as usually known p-mean) respectivelyOne can conclude that 119860119901 reduces to arithmetic mean andharmonic mean in the ordinary sense for 119901 = 1 and 119901 = minus1respectively
Remark 5 It is clear that Definition 4 can be written byusing various generators In particular if we take120573-arithmeticinstead of 120572-arithmetic then the mean can be defined by
119860120573 = 119899
120573sum119896=1
119909119896 (16)
Definition 6 (120572-geometric mean) Let 1199091 1199092 119909119899 isin R+The 120572-geometric mean namely 119866120572 is 119899th 120572-root of the 120572-product of (119909119899)rsquos
119866120572 = ( 119899
120572prod119896=1
119909119896)(1119899)120572 = 120572
( 119899prod119896=1
120572minus1 (119909119896))1119899
= 120572(120572minus1 (1199091) 120572minus1 (1199092) 120572minus1 (119909119899))1119899 (17)
We conclude similarly by taking the generators 120572 = expor 120572 = 119902119901 that the 120572-geometric mean can be interpreted asfollows
119866exp = exp (ln1199091 ln1199092 ln119909119899)1119899 (119909119899 gt 1)119866119901 = (1199091199011 1199091199012 119909119901119899)11198991119901 = (1199091 1199092 119909119899)1119899
(119901 = 0) (18)
119866exp and 119866119901 are called multiplicative geometric mean and119901-geometric mean respectively It would clearly have 119866119901 =119860exp for 119901 = 1Definition 7 (120572-harmonic mean) Let 1199091 1199092 119909119899 isin R+ and120572minus1(119909119899) = 0 for each 119899 isin N The 120572-harmonic mean 119867120572 isdefined by
119867120572 = 119899120572sum119896=1
(1 119909119896)= (1 1199091 + 1 1199092 + sdot sdot sdot + 1 119909119899)= 120572 1198991120572minus1 (1199091) + 1120572minus1 (1199092) + sdot sdot sdot + 1120572minus1 (119909119899)
(19)
Similarly one obtains that
119867exp
= exp 1198991ln (1199091) + 1ln (1199092) + sdot sdot sdot + 1ln (119909119899) (119909119899 gt 1)
119867119901 = 1198991 (1199091)119901 + 1 (1199092)119901 + sdot sdot sdot + 1 (119909119899)119901
1119901
(20)
119867exp and119867119901 are called multiplicative harmonic mean and 119901-harmonic mean respectively Obviously the inclusion (20) isreduced to ordinary harmonic mean and ordinary arithmeticmean for 119901 = 1 and 119901 = minus1 respectively
31 Non-Newtonian Weighted Means The weighted mean issimilar to an arithmetic mean where instead of each of thedata points contributing equally to the final average somedata points contributemore than othersMoreover the notionof weighted mean plays a role in descriptive statistics andalso occurs in a more general form in several other areas ofmathematics
The following definitions can give the relationshipsbetween the non-Newtonian weighted means and ordinaryweighted means
Definition 8 (weighted 120572-arithmetic mean) Formally theweighted 120572-arithmetic mean of a nonempty set of data1199091 1199092 119909119899 with nonnegative weights 1199081 1199082 119908119899 isthe quantity
International Journal of Analysis 5
120572 = 119899
120572sum119894=1
(119909119894 times 119894) 119899120572sum119894=1
119894 = (1199091 times 1 + 1199092 times 2 + sdot sdot sdot + 119909119899 times 119899) (1 + 2 + sdot sdot sdot + 119899) = 120572sum119899119894=1 119908119894120572minus1 (119909119894)sum119899119894=1 119908119894 (21)
The formulas are simplified when the weights are 120572-normalized such that they 120572-sum up to 120572sum119899119894=1119894 = 1 Forsuch normalized weights the weighted 120572-arithmetic mean issimply 120572 = 120572sum119899119894=1 119909119894 times 119894 Note that if all the weights areequal the weighted 120572-arithmetic mean is the same as the 120572-arithmetic mean
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-arithmeticmean can be given with the weights 1199081 1199082 119908119899 as fol-lows
exp = expsum119899119894=1 ln (119909119894) 119908119894sum119899119894=1 119908119894 = ( 119899prod119894=1
119909119908119894119894 )1sum119899119894=1 119908119894 (119909119899 gt 1)
119902119901 = 1199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)1199011199081 + 1199082 + sdot sdot sdot + 119908119899 1119901
(22)
exp and 119901 are called multiplicative weighted arithmeticmean and weighted 119901-arithmetic mean respectively expturns out to the ordinary weighted geometric mean Alsoone easily can see that 119901 is reduced to ordinary weightedarithmetic mean and weighted harmonic mean for 119901 = 1 and119901 = minus1 respectivelyDefinition 9 (weighted 120572-geometric mean) Given a set ofpositive reals 1199091 1199092 119909119899 and corresponding weights1199081 1199082 119908119899 then the weighted 120572-geometric mean 120572 isdefined by
120572 = 119899
120572prod119894=1
119909119894(119908119894sum
119899119894=1 119908119894)120572 = 119909(1199081(1199081+sdotsdotsdot+119908119899))1205721
times 119909(1199082(1199081+sdotsdotsdot+119908119899))1205722 times sdot sdot sdot times 119909(119908119899(1199081+sdotsdotsdot+119908119899))120572119899
= 120572[120572minus1 (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [120572minus1 (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [120572minus1 (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(23)
Note that if all the weights are equal the weighted 120572-geometric mean is the same as the 120572-geometric meanTaking 120572 = exp and 120572 = 119902119901 theweighted 120572-geometricmeancan be written for the weights 1199081 1199082 119908119899 as follows
exp = exp [ln (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [ln (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [ln (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(119909119899 gt 1) 119901 = 1199091199011199081(1199081+sdotsdotsdot+119908119899)1 1199091199011199082(1199081+sdotsdotsdot+119908119899)2 119909119901119908119899(1199081+sdotsdotsdot+119908119899)119899 1119901 = ( 119899prod
119894=1
119909119908119894119894 )1sum119899119894=1 119908119894
(24)
exp and 119901 are called weighted multiplicative geometricmean and weighted 119902-geometric mean Also we have 119901 =exp for all 119909119899 gt 1Definition 10 (weighted 120572-harmonic mean) If a set 11990811199082 119908119899 of weights is associated with the data set1199091 1199092 119909119899 then the weighted 120572-harmonic mean isdefined by
120572 = 119899
120572sum119896=1
119896 119899120572sum119896=1
(119896 119909119896)= (1 + sdot sdot sdot + 119899) (1 1199091 + 2 1199092 + sdot sdot sdot + 119899 119909119899)= 120572 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081120572minus1 (1199091) + 1199082120572minus1 (1199092) + sdot sdot sdot + 119908119899120572minus1 (119909119899)
(25)
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-harmonicmean with the weights 1199081 1199082 119908119899 can be written asfollows
exp
= exp 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 ln1199091 + 1199082 ln1199092 + sdot sdot sdot + 119908119899 ln119909119899 (119909119899 gt 1)
119901= 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)119901
1119901
(26)
exp and 119901 are called multiplicative weighted harmonicmean and weighted 119901-geometric mean respectively It isobvious that 119901 is reduced to ordinary weighted harmonicmean and ordinary weighted arithmetic mean for 119901 = 1 and119901 = minus1 respectively
6 International Journal of Analysis
Table 1 Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means
Weight Data 119901 119860119901 119866119901 119867119901 119901 119901 119901 119866exp 119867exp exp exp1199081 = 2 1199091 = 15 10 2750 2474 2264 3282 2988 2727 2402 2336 2905 28261199082 = 5 1199092 = 20 01 2499 2474 2450 3016 2988 2959 2402 2336 2905 28261199083 = 7 1199093 = 25 20 3061 2474 2114 3577 2988 2521 2402 2336 2905 28261199084 = 9 1199094 = 50 50 3822 2474 1873 4169 2988 2155 2402 2336 2905 2826
In Table 1 the non-Newtonian means are obtained byusing different generating functions For 120572 = 119902119901 the119901-means119860119901 119866119901 and119867119901 are reduced to ordinary arithmeticmean geo-metric mean and harmonic mean respectively In particularsome changes are observed for each value of 119860119901 119866119901 and119867119901means depending on the choice of 119901 As shown in the tablefor increasing values of 119901 the 119901-arithmetic mean 119860119901 and itsweighted form 119901 increase in particular 119901 tends to infin andthese means converge to the value of max119909119899 Converselyfor increasing values of 119901 the 119901-harmonic mean 119867119901 andits weighted forms 119901 decrease In particular these meansconverge to the value of min119909119899 as 119901 rarr infin Dependingon the choice of 119901 weighted forms 119901 and 119901 can beincreased or decreased without changing any weights Forthis reason this approach brings a new perspective to theconcept of classical (weighted) mean Moreover when wecompare 119867exp and ordinary harmonic mean in Table 1 wealso see that ordinary harmonic mean is smaller than 119867expOn the contrary119860exp and119866exp are smaller than their classicalforms 119860119901 and 119866119901 for 119901 = 1 Therefore we assert thatthe values of 119866exp 119867exp exp and exp should be evaluatedsatisfactorily
Corollary 11 Consider 119899 positive real numbers 1199091 1199092 119909119899Then the conditions119867120572 lt 119866120572 lt 119860120572 and 120572 lt 120572 lt 120572 holdwhen 120572 = exp for all 119909119899 gt 1 and 120572 = 119902119901 for all 119901 isin R+
4 Non-Newtonian Convexity
In this section the notion of non-Newtonian convex (lowast-convex) functions will be given by using different genera-tors Furthermore the relationships between lowast-convexity andnon-Newtonian weighted mean will be determined
Definition 12 (generalized lowast-convex function) Let 119868120572 be aninterval in R120572 Then 119891 119868120572 rarr R120573 is said to be lowast-convex if
119891 (1205821 times 119909 + 1205822 times 119910) le 1205831 times 119891 (119909) + 1205832 times 119891 (119910) (27)
holds where 1205821 + 1205822 = 1 and 1205831 + 1205832 = 1 for all 1205821 1205822 isin[0 1] and 1205831 1205832 isin [0 1] Therefore by combining this withthe generators 120572 and 120573 we deduce that
119891 (120572 120572minus1 (1205821) 120572minus1 (119909) + 120572minus1 (1205822) 120572minus1 (119910))le 120573 120573minus1 (1205831) 120573minus1119891 (119909) + 120573minus1 (1205832) 120573minus1119891 (119910) (28)
If (28) is strict for all 119909 = 119910 then 119891 is said to be strictly lowast-convex If the inequality in (28) is reversed then 119891 is said to
be lowast-concave On the other hand the inclusion (28) can bewritten with respect to the weighted 120572-arithmetic mean in(21) as follows
119891 (120572 119909 119910) le 120573 119891 (119909) 119891 (119910) (29)
Remark 13 We remark that the definition of lowast-convexity in(27) can be evaluated by non-Newtonian coordinate systeminvolving lowast-lines (see Definition 3) For 120572 = 120573 = 119868 the lowast-lines are straight lines in two-dimensional Euclidean spaceFor this reason we say that almost all the properties ofordinary Cartesian coordinate system will be valid for non-Newtonian coordinate system under lowast-arithmetic
Also depending on the choice of generator functionsthe definition of lowast-convexity in (27) can be interpreted asfollows
Case 1 (a) If we take 120572 = 120573 = exp and 1205821 = 1205831 1205822 = 1205832 in(28) then
119891 (119909ln1205821119910ln1205822) le 119891 (119909)ln1205821 119891 (119910)ln1205822 (1205821 1205822 isin [1 119890])
(30)
where 12058211205822 = 119890 holds and 119891 119868exp rarr Rexp = (0infin) iscalled bigeometric (usually known as multiplicative) convexfunction (cf [2]) Equivalently 119891 is bigeometric convex ifand only if log119891(119909) is an ordinary convex function
(b) For 120572 = exp and 120573 = 119868 we have119891 (119909ln1205821119910ln1205822) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 isin [1 119890] 1205831 1205832 isin [0 1]) (31)
where 12058211205822 = 119890 and 1205831 +1205832 = 1 In this case the function 119891 119868exp rarr R is called geometric convex function Everygeometric convex (usually known as log-convex) function isalso convex (cf [2])
(c) Taking 120572 = 119868 and 120573 = exp one obtains
119891 (1205821119909 + 1205822119910) le 119891 (119909)ln 1205831 119891 (119910)ln 1205832 (1205831 1205832 isin [1 119890] 1205821 1205822 isin [0 1]) (32)
where 12058311205832 = 119890 and 1205821 + 1205822 = 1 and 119891 119868 rarr Rexp is calledanageometric convex function
Case 2 (a) If 120572 = 120573 = 119902119901 in (28) then
119891(((1205821119909)119901 + (1205822119910)119901)1119901)le ((1205821119891 (119909))119901 + (1205822119891 (119910))119901)1119901 (119901 isin R
+) (33)
International Journal of Analysis 7
where 1205821 1205822 isin [0 1] 1205821199011 + 1205821199012 = 1 and 119891 119868119902119901 rarr R119902119901 iscalled 119876119876-convex function
(b) For 120572 = 119902119901 and 120573 = 119868 we write that119891(((1205821119909)119901 + (1205822119910)119901)1119901) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 1205831 1205832 isin [0 1]) (34)
where 1205821199011 + 1205821199012 = 1 1205831 + 1205832 = 1 and 119891 119868119902119901 rarr R is called119876119868-convex function(c) For 120572 = 119868 and 120573 = 119902119901 we obtain that
119891 (1205821119909 + 1205822119910) le ((1205831119891 (119909))119901 + (1205832119891 (119910))119901)1119901 (1205821 1205822 1205831 1205832 isin [0 1])
(35)
where 1205831199011 +1205831199012 = 1 1205821 +1205822 = 1 and 119891 119868 rarr R119902119901 is called 119868119876-convex function
The lowast-convexity of a function 119891 119868120572 rarr R120573 means geo-metrically that the lowast-points of the graph of 119891 are under thechord joining the endpoints (119886 119891(119886)) and (119887 119891(119887)) on non-Newtonian coordinate system for every 119886 119887 isin 119868120572 By takinginto account the definition of lowast-slope in Definition 3 we have
(119891 (119909) minus 119891 (119886)) 120580 (119909 minus 119886)le (119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886) (36)
which implies
119891 (119909) le 119891 (119886) + ((119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886))times 120580 (119909 minus 119886) (37)
for all 119909 isin [ ]On the other hand (37) means that if 119875119876 and 119877 are any
three lowast-points on the graph of 119891 with 119876 between 119875 and 119877then 119876 is on or below chord 119875119877 In terms of lowast-slope it isequivalent to
119898lowast (119875119876) le119898lowast (119875119877) le119898lowast (119876119877) (38)
with strict inequalities when 119891 is strictly lowast-convexNow to avoid the repetition of the similar statements we
give some necessary theorems and lemmas
Lemma 14 (Jensenrsquos inequality) A 120573-real-valued function 119891defined on an interval 119868120572 is lowast-convex if and only if
119891( 119899
120572sum119896=1
120582119896 times 119909119896) le 119899
120573sum119896=1
120583119896 times 119891 (119909119896) (39)
holds where 120572sum119899119896=1 120582119896 = 1 and 120573sum119899119896=1 120583119896 = 1 for all 120582119899 isin [0 1]and 120583119899 isin [0 1]Proof The proof is straightforward hence omitted
Theorem 15 Let 119891 119868120572 rarr R120573 be a lowast-continuous functionThen 119891 is lowast-convex if and only if 119891 is midpoint lowast-convexthat is
1199091 1199092 isin 119868120572 implies
119891 (119860120572 1199091 1199092) le 119860120573 119891 (1199091) 119891 (1199092) (40)
Proof The proof can be easily obtained using the inequality(39) in Lemma 14
Theorem16 (cf [2]) Let 119891 119868exp rarr Rexp be alowast-differentiablefunction (see [19]) on a subinterval 119868exp sube (0infin) Then thefollowing assertions are equivalent
(i) 119891 is bigeometric convex (concave)(ii) The function 119891lowast(119909) is increasing (decreasing)
Corollary 17 A positive 120573-real-valued function 119891 defined onan interval 119868exp is bigeometric convex if and only if
119891 (119909ln12058211 119909ln12058222 119909ln120582119899119899 )le 119891 (1199091)ln1205821 119891 (1199092)ln1205822 119891 (119909119899)ln120582119899
(41)
holds where prod119899119896=1120582119896 = 119890 for all 1199091 1199092 119909119899 isin 119868exp and1205821 1205822 120582119899 isin [1 119890] Besides we have119891 (exp 1199091 1199092 119909119899)le exp 119891 (1199091) 119891 (1199092) 119891 (119909119899) (42)
Corollary 18 A 120573-real-valued function119891 defined on an inter-val 119868119902119901 is 119876119876-convex if and only if
119891[((12058211199091)119901 + (12058221199092)119901 + sdot sdot sdot + (120582119899119909119899)119901)1119901]le ((1205821119891 (1199091))119901 + (1205822119891 (1199092))119901 + sdot sdot sdot+ (120582119899119891 (119909119899))119901)1119901
(43)
holds where sum119899119896=1 120582119901119896 = 1 for all 1199091 1199092 119909119899 isin 119868119902119901 and1205821 1205822 120582119899 isin [0 1] Thus we have
119891 (119901 1199091 1199092 119909119899)le 119901 119891 (1199091) 119891 (1199092) 119891 (119909119899) (119901 isin R
+) (44)
5 An Application of Multiplicative Continuity
In this section based on the definition of bigeometric convexfunction and multiplicative continuity we get an analogue ofordinary Lipschitz condition on any closed interval
Let 119891 be a bigeometric (multiplicative) convex functionand finite on a closed interval [119909 119910] sub R+ It is obvious that119891is bounded from above by 119872 = max119891(119909) 119891(119910) sincefor any 119911 = 1199091205821199101minus120582 in the interval 119891(119911) le 119891(119909)120582119891(119910)1minus120582 for
8 International Journal of Analysis
120582 isin [1 119890] It is also bounded from below as we see by writingan arbitrary point in the form 119905radic119909119910 for 119905 isin R+ Then
1198912 (radic119909119910) le 119891 (119905radic119909119910)119891(radic119909119910119905 ) (45)
Using 119872 as the upper bound 119891(radic119909119910119905) we obtain119891 (119905radic119909119910) ge 11198721198912 (radic119909119910) = 119898 (46)
Thus a bigeometric convex functionmay not be continuous atthe boundary points of its domainWewill prove that for anyclosed subinterval [119909 119910] of the interior of the domain thereis a constant 119870 gt 0 so that for any two points 119886 119887 isin [119909 119910] subR+
119891 (119886)119891 (119887) le (119886119887)119870 (47)
A function that satisfies (47) for some 119870 and all 119886 and 119887 inan interval is said to satisfy bigeometric Lipschitz conditionon the interval
Theorem 19 Suppose that 119891 119868 rarr R+ is multiplicative con-vex Then 119891 satisfies the multiplicative Lipschitz condition onany closed interval [119909 119910] sub R+ contained in the interior 1198680 of119868 that is 119891 is continuous on 1198680Proof Take 120576 gt 1 so that [119909120576 119910120576] isin 119868 and let 119898 and 119872 bethe lower and upper bounds for 119891 on [119909120576 119910120576] If 119903 and 119904 aredistinct points of [119909 119910] with 119904 gt 119903 set
119911 = 119904120576120582 = ( 119904119903)
1 ln(120576119904119903) (120582 isin (1 119890)) (48)
Then 119911 isin [119909120576 119910120576] and 119904 = 119911ln1205821199031minusln120582 and we obtain
119891 (119904) le 119891 (119911)ln120582 119891 (119903)1minusln120582 = (119891 (119911)119891 (119903) )ln120582 119891 (119903) (49)
which yields
ln(119891 (119904)119891 (119903)) le ln 120582 ln(119891 (119911)119891 (119903) )lt ln( 119904119903)
1 ln(120576)ln(119872119898 )
119891 (119904)119891 (119903) le ( 119904119903)ln(119872119898) ln(120576)
(50)
where 119870 = ln(119872119898) ln(120576) gt 0 Since the points 119903 119904 isin [119909 119910]are arbitrary we get 119891 that satisfies a multiplicative Lipschitzcondition The remaining part can be obtained in the similarway by taking 119904 lt 119903 and 119911 = 119904120576 Finally 119891 is continuoussince [119909 119910] is arbitrary in 1198680
6 Concluding Remarks
Although all arithmetics are isomorphic only by distinguish-ing among them do we obtain suitable tools for construct-ing all the non-Newtonian calculi But the usefulness ofarithmetic is not limited to the construction of calculi webelieve there is a more fundamental reason for consideringalternative arithmetics theymay also be helpful in developingand understanding new systems of measurement that couldyield simpler physical laws
In this paper it was shown that due to the choice ofgenerator function 119860119901 119866119901 and 119867119901 means are reduced toordinary arithmetic geometric and harmonic mean respec-tively As shown in Table 1 for increasing values of 119901 119860119901and 119901 means increase especially 119901 rarr infin these meansconverge to the value of max119909119899 Conversely for increasingvalues of 119901 119867119901 and 119901 means decrease especially 119901 rarr infinthese means converge to the value of min119909119899 Additionallywe give some new definitions regarding convex functionswhich are plotted on the non-Newtonian coordinate systemObviously for different generator functions one can obtainsome new geometrical interpretations of convex functionsOur future works will include the most famous HermiteHadamard inequality for the class of lowast-convex functionsCompeting Interests
The authors declare that they have no competing interests
References
[1] C Niculescu and L-E Persson Convex Functions and TheirApplications A Contemporary Approach Springer Berlin Ger-many 2006
[2] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000
[3] R Webster Convexity Oxford University Press New York NYUSA 1995
[4] J Banas and A Ben Amar ldquoMeasures of noncompactness inlocally convex spaces and fixed point theory for the sum oftwo operators on unbounded convex setsrdquo CommentationesMathematicae Universitatis Carolinae vol 54 no 1 pp 21ndash402013
[5] J Matkowski ldquoGeneralized weighted and quasi-arithmeticmeansrdquo Aequationes Mathematicae vol 79 no 3 pp 203ndash2122010
[6] D Głazowska and J Matkowski ldquoAn invariance of geometricmean with respect to Lagrangian meansrdquo Journal of Mathemat-ical Analysis and Applications vol 331 no 2 pp 1187ndash1199 2007
[7] J Matkowski ldquoGeneralized weighted quasi-arithmetic meansand the Kolmogorov-Nagumo theoremrdquo Colloquium Mathe-maticum vol 133 no 1 pp 35ndash49 2013
[8] N Merentes and K Nikodem ldquoRemarks on strongly convexfunctionsrdquo Aequationes Mathematicae vol 80 no 1-2 pp 193ndash199 2010
[9] M Grossman Bigeometric Calculus Archimedes FoundationBox 240 Rockport Mass USA 1983
[10] M Grossman and R Katz Non-Newtonian Calculus NewtonInstitute 1978
International Journal of Analysis 9
[11] M Grossman The First Nonlinear System of Differential andIntegral Calculus 1979
[12] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008
[13] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007
[14] E Mısırlı and Y Gurefe ldquoMultiplicative adams-bashforth-moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011
[15] A F Cakmak and F Basar ldquoSome new results on sequencespaces with respect to non-Newtonian calculusrdquo Journal ofInequalities and Applications vol 2012 article 228 2012
[16] A F Cakmak and F Basar ldquoCertain spaces of functions overthe field of non-Newtonian complex numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 236124 12 pages 2014
[17] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013
[18] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014
[19] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015
[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014
[21] U Kadak ldquoNon-Newtonian fuzzy numbers and related applica-tionsrdquo Iranian Journal of Fuzzy Systems vol 12 no 5 pp 117ndash1372015
[22] Y Gurefe U Kadak E Misirli and A Kurdi ldquoA new look atthe classical sequence spaces by using multiplicative calculusrdquoUniversity Politehnica of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 78 no 2 pp 9ndash20 2016
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Stochastic AnalysisInternational Journal of
2 International Journal of Analysis
In Section 3 first the definitions of non-Newtonian meansare given which will be used for non-Newtonian convexity Inthis section the forms of weighted means are presented andan illustrative table is given In Section 4 the generalized non-Newtonian convex function is defined on the interval 119868120572 andsome types of convex function are obtained by using differentgenerators In the final section of the paper we assert thenotion of multiplicative Lipschitz condition on the closedinterval [119909 119910] sub (0infin)2 Preliminary Background and Notation
Arithmetic is any system that satisfies the whole of theordered field axioms whose domain is a subset of R Thereare infinitely many types of arithmetic all of which areisomorphic that is structurally equivalent
A generator 120572 is a one-to-one function whose domain isR and whose range is a subset R120572 of R where R120572 = 120572(119909) 119909 isin R Each generator generates exactly one arithmeticand conversely each arithmetic is generated by exactly onegenerator If 119868(119909) = 119909 for all 119909 isin R the identity functionrsquosinverse is itself In the special cases 120572 = 119868 and 120572 = exp 120572generates the classical and geometric arithmetic respectivelyBy 120572-arithmetic we mean the arithmetic whose domain is Rand whose operations are defined as follows for 119909 119910 isin R120572and any generator 120572
120572-addition119909 + 119910 = 120572 120572minus1 (119909) + 120572minus1 (119910) 120572-subtraction119909 minus 119910 = 120572 120572minus1 (119909) minus 120572minus1 (119910)
120572-multiplication119909 times 119910 = 120572 120572minus1 (119909) times 120572minus1 (119910) 120572-division 119909 119910 = 120572 120572minus1 (119909) divide 120572minus1 (119910)
120572-order 119909 lt 119910 lArrrArr 120572minus1 (119909) lt 120572minus1 (119910)
(1)
As a generator we choose exp function acting from R intothe set Rexp = (0infin) as follows
120572 R 997888rarr Rexp
119909 997891997888rarr 119910 = 120572 (119909) = 119890119909 (2)
It is obvious that 120572-arithmetic reduces to the geometric arith-metic as follows
geometric addition 119909 + 119910 = 119890ln119909+ln119910 = 119909 sdot 119910geometric subtraction 119909 minus 119910 = 119890ln119909minusln119910 = 119909 divide 119910geometric multiplication 119909 times 119910 = 119890ln 119909 ln119910 = 119909ln119910= 119910ln119909
geometric division 119909 119910 = 119890ln 119909 ln119910 = 1199091 ln119910geometric order 119909 lt 119910 lArrrArr ln (119909) lt ln (119910)
(3)
Following Grossman and Katz [10] we give the infinitelymany 119902119901-arithmetics of which the quadratic and harmonic
arithmetic are special cases for119901 = 2 and119901 = minus1 respectivelyThe function 119902119901 Rrarr R119902 sube R and its inverse 119902minus1119901 are definedas follows (119901 isin R 0)
119902119901 (119909) =
1199091119901 119909 gt 00 119909 = 0minus (minus119909)1119901 119909 lt 0
119902minus1119901 (119909) =
119909119901 119909 gt 00 119909 = 0minus (minus119909)119901 119909 lt 0
(4)
It is to be noted that 119902119901-calculus is reduced to the classicalcalculus for 119901 = 1 Additionally it is concluded that the 120572-summation can be given as follows
119899
120572sum119896=1
119909119896 = 120572 119899sum119896=1
120572minus1 (119909119896)= 120572 120572minus1 (1199091) + sdot sdot sdot + 120572minus1 (119909119899) forall119909119896 isin R
+(5)
Definition 1 (see [15]) Let119883 = (119883 119889120572) be an 120572-metric spaceThen the basic notions can be defined as follows
(a) A sequence 119909 = (119909119896) is a function from the set N intothe setR120572 The 120572-real number 119909119896 denotes the value ofthe function at 119896 isin N and is called the 119896th term of thesequence
(b) A sequence (119909119899) in 119883 = (119883 119889120572) is said to be 120572-convergent if for every given 120576 gt 0 (120576 isin R120572) thereexist an 1198990 = 1198990(120576) isin N and 119909 isin 119883 such that119889120572(119909119899 119909) = |119909119899 minus 119909|120572 lt 120576 for all 119899 gt 1198990 and is denotedby 120572lim119899rarrinfin119909119899 = 119909 or 119909119899 120572997888rarr 119909 as 119899 rarr infin
(c) A sequence (119909119899) in119883 = (119883 119889120572) is said to be 120572-Cauchyif for every 120576 gt 0 there is an 1198990 = 1198990(120576) isin N such that119889120572(119909119899 119909119898) lt 120576 for all119898 119899 gt 1198990
Throughout this paper we define the 119901th 120572-exponent 119909119901120572and 119902th 120572-root 119909(1119902)120572 of 119909 isin R+ by
1199092120572 = 119909 times 119909 = 120572 120572minus1 (119909) times 120572minus1 (119909)= 120572 [120572minus1 (119909)]2
1199093120572 = 1199092120572 times 119909= 120572 120572minus1 120572 [120572minus1 (119909) times 120572minus1 (119909)] times 120572minus1 (119909)= 120572 [120572minus1 (119909)]3
119909119901120572 = 119909(119901minus1)120572 times 119909 = 120572 [120572minus1 (119909)]119901
(6)
International Journal of Analysis 3
and 120572radic119909 = 119909(12)120572 = 119910 provided there exists an 119910 isin R120572 suchthat 1199102120572 = 11990921 lowast-Arithmetic Suppose that 120572 and 120573 are two arbitrarilyselected generators and (ldquostar-rdquo) also is the ordered pairof arithmetics (120573-arithmetic and 120572-arithmetic) The sets(R120573 + minus times lt) and (R120572 + minus times lt) are complete orderedfields and 119887119890119905119886(119886119897119901ℎ119886)-generator generates 119887119890119905119886(119886119897119901ℎ119886)-arithmetic respectively Definitions given for 120573-arithmeticare also valid for 120572-arithmetic Also 120572-arithmetic is used forarguments and 120573-arithmetic is used for values in particu-lar changes in arguments and values are measured by 120572-differences and 120573-differences respectively
Let 119909 isin (R120572 + minus times lt) and 119910 isin (R120573 + minus times lt) bearbitrarily chosen elements from corresponding arithmeticThen the ordered pair (119909 119910) is called a lowast-point and the setof all lowast-points is called the set of lowast-complex numbers and isdenoted by Clowast that is
Clowast fl 119911lowast = (119909 119910) | 119909 isin R120572 119910 isin R120573 (7)
Definition 2 (see [17]) (a) The lowast-limit of a function 119891 at anelement 119886 inR120572 is if it exists the unique number 119887 inR120573 suchthatlowast lim119909rarr119886
119891 (119909) = 119887 lArrrArrforall120576 gt 0 exist120575 gt 0 ni 1003816100381610038161003816119891 (119909) minus 1198871003816100381610038161003816120573 lt 120576
forall119909 120576 isin R120572 1003816100381610038161003816119909 minus 1198861003816100381610038161003816120572 lt 120575(8)
for 120575 isin R120573 and is written as lowastlim119909rarr119886119891(119909) = 119887A function119891 islowast-continuous at a point 119886 inR120572 if and only
if 119886 is an argument of 119891 and lowastlim119909rarr119886119891(119909) = 119891(119886) When120572 and 120573 are the identity function 119868 the concepts of lowast-limitand lowast-continuity are reduced to those of classical limit andclassical continuity
(b) The isomorphism from 120572-arithmetic to 120573-arithmeticis the unique function 120580 (iota) which has the following threeproperties
(i) 120580 is one to one(ii) 120580 is from R120572 to R120573
(iii) For any numbers 119906 V isin R120572
120580 (119906 + V) = 120580 (119906) + 120580 (V) 120580 (119906 minus V) = 120580 (119906) minus 120580 (V) 120580 (119906 times V) = 120580 (119906) times 120580 (V) 120580 (119906 V) = 120580 (119906) 120580 (V)
(9)
It turns out that 120580(119909) = 120573120572minus1(119909) for every 119909 in R120572 andthat 120580() = for every 120572-integer Since for example119906 + V = 120580minus1120580(119906) + 120580(V) it should be clear that any statement in
120572-arithmetic can readily be transformed into a statement in120573-arithmetic
Definition 3 (see [10]) The following statements are valid
(i) The lowast-points 1198751 1198752 and 1198753 are lowast-collinear providedthat at least one of the following holds
119889lowast (1198752 1198751) + 119889lowast (1198751 1198753) = 119889lowast (1198752 1198753) 119889lowast (1198751 1198752) + 119889lowast (1198752 1198753) = 119889lowast (1198751 1198753) 119889lowast (1198751 1198753) + 119889lowast (1198753 1198752) = 119889lowast (1198751 1198752)
(10)
(ii) A lowast-line is a set 119871 of at least two distinct points suchthat for any distinct points1198751 and1198752 in 119871 a point1198753 isin 119871 if and only if 1198751 1198752 and 1198753 are lowast-collinear When120572 = 120573 = 119868 the lowast-lines are the straight lines in two-dimensional Euclidean space
(iii) The lowast-slope of a lowast-line through the points (1198861 1198871) and(1198862 1198872) is given by
119898lowast = (1198872 minus 1198871) 120580 (1198862 minus 1198861)= 120573 120573minus1 (1198872) minus 120573minus1 (1198871)120572minus1 (1198862) minus 120572minus1 (1198861) (1198861 = 1198862) (11)
for 1198861 1198862 isin R120572 and 1198871 1198872 isin R120573
If the following lowast-limit in (12) exists we denote it by119891lowast(119905) call it the lowast-derivative of 119891 at 119905 and say that 119891 is lowast-differentiable at 119905 (see [19])
lowastlim119909rarr119905
(119891 (119909) minus 119891 (119905)) 120580 (119909 minus 119905)= lim119909rarr119905
120573120573minus1 119891 (119909) minus 120573minus1 119891 (119905)120572minus1 (119909) minus 120572minus1 (119905)
= lim119909rarr119905
120573120573minus1 119891 (119909) minus 120573minus1 119891 (119905)119909 minus 119905sdot 119909 minus 119905120572minus1 (119909) minus 120572minus1 (119905) = 120573
(120573minus1 ∘ 119891)1015840 (119905)(120572minus1)1015840 (119905)
(12)
3 Non-Newtonian (Weighted) Means
Definition 4 (120572-arithmetic mean) Consider that 119899 positivereal numbers 1199091 1199092 119909119899 are given The 120572-mean (average)
4 International Journal of Analysis
denoted by 119860120572 is the 120572-sum of 119909119899rsquos 120572-divided by for all119899 isin N That is
119860120572 = 119899
120572sum119896=1
119909119896 = 119899
120572sum119896=1
120572120572minus1 (119909119896)119899
= 120572120572minus1 (1199091) + 120572minus1 (1199092) + sdot sdot sdot + 120572minus1 (119909119899)119899 (13)
For 120572 = exp we obtain that
119860exp = ( 119899prod119896=1
119909119896)1119899
= (1199091 sdot 1199092 sdot sdot sdot 119909119899)1119899 (14)
Similarly for 120572 = 119902119901 we get
119860119901 = (1199091199011 + 1199091199012 + sdot sdot sdot + 119909119901119899119899 )1119901 119901 isin R 0 (15)
119860exp and 119860119901 are called multiplicative arithmetic mean and119901-arithmetic mean (as usually known p-mean) respectivelyOne can conclude that 119860119901 reduces to arithmetic mean andharmonic mean in the ordinary sense for 119901 = 1 and 119901 = minus1respectively
Remark 5 It is clear that Definition 4 can be written byusing various generators In particular if we take120573-arithmeticinstead of 120572-arithmetic then the mean can be defined by
119860120573 = 119899
120573sum119896=1
119909119896 (16)
Definition 6 (120572-geometric mean) Let 1199091 1199092 119909119899 isin R+The 120572-geometric mean namely 119866120572 is 119899th 120572-root of the 120572-product of (119909119899)rsquos
119866120572 = ( 119899
120572prod119896=1
119909119896)(1119899)120572 = 120572
( 119899prod119896=1
120572minus1 (119909119896))1119899
= 120572(120572minus1 (1199091) 120572minus1 (1199092) 120572minus1 (119909119899))1119899 (17)
We conclude similarly by taking the generators 120572 = expor 120572 = 119902119901 that the 120572-geometric mean can be interpreted asfollows
119866exp = exp (ln1199091 ln1199092 ln119909119899)1119899 (119909119899 gt 1)119866119901 = (1199091199011 1199091199012 119909119901119899)11198991119901 = (1199091 1199092 119909119899)1119899
(119901 = 0) (18)
119866exp and 119866119901 are called multiplicative geometric mean and119901-geometric mean respectively It would clearly have 119866119901 =119860exp for 119901 = 1Definition 7 (120572-harmonic mean) Let 1199091 1199092 119909119899 isin R+ and120572minus1(119909119899) = 0 for each 119899 isin N The 120572-harmonic mean 119867120572 isdefined by
119867120572 = 119899120572sum119896=1
(1 119909119896)= (1 1199091 + 1 1199092 + sdot sdot sdot + 1 119909119899)= 120572 1198991120572minus1 (1199091) + 1120572minus1 (1199092) + sdot sdot sdot + 1120572minus1 (119909119899)
(19)
Similarly one obtains that
119867exp
= exp 1198991ln (1199091) + 1ln (1199092) + sdot sdot sdot + 1ln (119909119899) (119909119899 gt 1)
119867119901 = 1198991 (1199091)119901 + 1 (1199092)119901 + sdot sdot sdot + 1 (119909119899)119901
1119901
(20)
119867exp and119867119901 are called multiplicative harmonic mean and 119901-harmonic mean respectively Obviously the inclusion (20) isreduced to ordinary harmonic mean and ordinary arithmeticmean for 119901 = 1 and 119901 = minus1 respectively
31 Non-Newtonian Weighted Means The weighted mean issimilar to an arithmetic mean where instead of each of thedata points contributing equally to the final average somedata points contributemore than othersMoreover the notionof weighted mean plays a role in descriptive statistics andalso occurs in a more general form in several other areas ofmathematics
The following definitions can give the relationshipsbetween the non-Newtonian weighted means and ordinaryweighted means
Definition 8 (weighted 120572-arithmetic mean) Formally theweighted 120572-arithmetic mean of a nonempty set of data1199091 1199092 119909119899 with nonnegative weights 1199081 1199082 119908119899 isthe quantity
International Journal of Analysis 5
120572 = 119899
120572sum119894=1
(119909119894 times 119894) 119899120572sum119894=1
119894 = (1199091 times 1 + 1199092 times 2 + sdot sdot sdot + 119909119899 times 119899) (1 + 2 + sdot sdot sdot + 119899) = 120572sum119899119894=1 119908119894120572minus1 (119909119894)sum119899119894=1 119908119894 (21)
The formulas are simplified when the weights are 120572-normalized such that they 120572-sum up to 120572sum119899119894=1119894 = 1 Forsuch normalized weights the weighted 120572-arithmetic mean issimply 120572 = 120572sum119899119894=1 119909119894 times 119894 Note that if all the weights areequal the weighted 120572-arithmetic mean is the same as the 120572-arithmetic mean
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-arithmeticmean can be given with the weights 1199081 1199082 119908119899 as fol-lows
exp = expsum119899119894=1 ln (119909119894) 119908119894sum119899119894=1 119908119894 = ( 119899prod119894=1
119909119908119894119894 )1sum119899119894=1 119908119894 (119909119899 gt 1)
119902119901 = 1199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)1199011199081 + 1199082 + sdot sdot sdot + 119908119899 1119901
(22)
exp and 119901 are called multiplicative weighted arithmeticmean and weighted 119901-arithmetic mean respectively expturns out to the ordinary weighted geometric mean Alsoone easily can see that 119901 is reduced to ordinary weightedarithmetic mean and weighted harmonic mean for 119901 = 1 and119901 = minus1 respectivelyDefinition 9 (weighted 120572-geometric mean) Given a set ofpositive reals 1199091 1199092 119909119899 and corresponding weights1199081 1199082 119908119899 then the weighted 120572-geometric mean 120572 isdefined by
120572 = 119899
120572prod119894=1
119909119894(119908119894sum
119899119894=1 119908119894)120572 = 119909(1199081(1199081+sdotsdotsdot+119908119899))1205721
times 119909(1199082(1199081+sdotsdotsdot+119908119899))1205722 times sdot sdot sdot times 119909(119908119899(1199081+sdotsdotsdot+119908119899))120572119899
= 120572[120572minus1 (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [120572minus1 (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [120572minus1 (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(23)
Note that if all the weights are equal the weighted 120572-geometric mean is the same as the 120572-geometric meanTaking 120572 = exp and 120572 = 119902119901 theweighted 120572-geometricmeancan be written for the weights 1199081 1199082 119908119899 as follows
exp = exp [ln (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [ln (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [ln (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(119909119899 gt 1) 119901 = 1199091199011199081(1199081+sdotsdotsdot+119908119899)1 1199091199011199082(1199081+sdotsdotsdot+119908119899)2 119909119901119908119899(1199081+sdotsdotsdot+119908119899)119899 1119901 = ( 119899prod
119894=1
119909119908119894119894 )1sum119899119894=1 119908119894
(24)
exp and 119901 are called weighted multiplicative geometricmean and weighted 119902-geometric mean Also we have 119901 =exp for all 119909119899 gt 1Definition 10 (weighted 120572-harmonic mean) If a set 11990811199082 119908119899 of weights is associated with the data set1199091 1199092 119909119899 then the weighted 120572-harmonic mean isdefined by
120572 = 119899
120572sum119896=1
119896 119899120572sum119896=1
(119896 119909119896)= (1 + sdot sdot sdot + 119899) (1 1199091 + 2 1199092 + sdot sdot sdot + 119899 119909119899)= 120572 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081120572minus1 (1199091) + 1199082120572minus1 (1199092) + sdot sdot sdot + 119908119899120572minus1 (119909119899)
(25)
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-harmonicmean with the weights 1199081 1199082 119908119899 can be written asfollows
exp
= exp 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 ln1199091 + 1199082 ln1199092 + sdot sdot sdot + 119908119899 ln119909119899 (119909119899 gt 1)
119901= 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)119901
1119901
(26)
exp and 119901 are called multiplicative weighted harmonicmean and weighted 119901-geometric mean respectively It isobvious that 119901 is reduced to ordinary weighted harmonicmean and ordinary weighted arithmetic mean for 119901 = 1 and119901 = minus1 respectively
6 International Journal of Analysis
Table 1 Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means
Weight Data 119901 119860119901 119866119901 119867119901 119901 119901 119901 119866exp 119867exp exp exp1199081 = 2 1199091 = 15 10 2750 2474 2264 3282 2988 2727 2402 2336 2905 28261199082 = 5 1199092 = 20 01 2499 2474 2450 3016 2988 2959 2402 2336 2905 28261199083 = 7 1199093 = 25 20 3061 2474 2114 3577 2988 2521 2402 2336 2905 28261199084 = 9 1199094 = 50 50 3822 2474 1873 4169 2988 2155 2402 2336 2905 2826
In Table 1 the non-Newtonian means are obtained byusing different generating functions For 120572 = 119902119901 the119901-means119860119901 119866119901 and119867119901 are reduced to ordinary arithmeticmean geo-metric mean and harmonic mean respectively In particularsome changes are observed for each value of 119860119901 119866119901 and119867119901means depending on the choice of 119901 As shown in the tablefor increasing values of 119901 the 119901-arithmetic mean 119860119901 and itsweighted form 119901 increase in particular 119901 tends to infin andthese means converge to the value of max119909119899 Converselyfor increasing values of 119901 the 119901-harmonic mean 119867119901 andits weighted forms 119901 decrease In particular these meansconverge to the value of min119909119899 as 119901 rarr infin Dependingon the choice of 119901 weighted forms 119901 and 119901 can beincreased or decreased without changing any weights Forthis reason this approach brings a new perspective to theconcept of classical (weighted) mean Moreover when wecompare 119867exp and ordinary harmonic mean in Table 1 wealso see that ordinary harmonic mean is smaller than 119867expOn the contrary119860exp and119866exp are smaller than their classicalforms 119860119901 and 119866119901 for 119901 = 1 Therefore we assert thatthe values of 119866exp 119867exp exp and exp should be evaluatedsatisfactorily
Corollary 11 Consider 119899 positive real numbers 1199091 1199092 119909119899Then the conditions119867120572 lt 119866120572 lt 119860120572 and 120572 lt 120572 lt 120572 holdwhen 120572 = exp for all 119909119899 gt 1 and 120572 = 119902119901 for all 119901 isin R+
4 Non-Newtonian Convexity
In this section the notion of non-Newtonian convex (lowast-convex) functions will be given by using different genera-tors Furthermore the relationships between lowast-convexity andnon-Newtonian weighted mean will be determined
Definition 12 (generalized lowast-convex function) Let 119868120572 be aninterval in R120572 Then 119891 119868120572 rarr R120573 is said to be lowast-convex if
119891 (1205821 times 119909 + 1205822 times 119910) le 1205831 times 119891 (119909) + 1205832 times 119891 (119910) (27)
holds where 1205821 + 1205822 = 1 and 1205831 + 1205832 = 1 for all 1205821 1205822 isin[0 1] and 1205831 1205832 isin [0 1] Therefore by combining this withthe generators 120572 and 120573 we deduce that
119891 (120572 120572minus1 (1205821) 120572minus1 (119909) + 120572minus1 (1205822) 120572minus1 (119910))le 120573 120573minus1 (1205831) 120573minus1119891 (119909) + 120573minus1 (1205832) 120573minus1119891 (119910) (28)
If (28) is strict for all 119909 = 119910 then 119891 is said to be strictly lowast-convex If the inequality in (28) is reversed then 119891 is said to
be lowast-concave On the other hand the inclusion (28) can bewritten with respect to the weighted 120572-arithmetic mean in(21) as follows
119891 (120572 119909 119910) le 120573 119891 (119909) 119891 (119910) (29)
Remark 13 We remark that the definition of lowast-convexity in(27) can be evaluated by non-Newtonian coordinate systeminvolving lowast-lines (see Definition 3) For 120572 = 120573 = 119868 the lowast-lines are straight lines in two-dimensional Euclidean spaceFor this reason we say that almost all the properties ofordinary Cartesian coordinate system will be valid for non-Newtonian coordinate system under lowast-arithmetic
Also depending on the choice of generator functionsthe definition of lowast-convexity in (27) can be interpreted asfollows
Case 1 (a) If we take 120572 = 120573 = exp and 1205821 = 1205831 1205822 = 1205832 in(28) then
119891 (119909ln1205821119910ln1205822) le 119891 (119909)ln1205821 119891 (119910)ln1205822 (1205821 1205822 isin [1 119890])
(30)
where 12058211205822 = 119890 holds and 119891 119868exp rarr Rexp = (0infin) iscalled bigeometric (usually known as multiplicative) convexfunction (cf [2]) Equivalently 119891 is bigeometric convex ifand only if log119891(119909) is an ordinary convex function
(b) For 120572 = exp and 120573 = 119868 we have119891 (119909ln1205821119910ln1205822) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 isin [1 119890] 1205831 1205832 isin [0 1]) (31)
where 12058211205822 = 119890 and 1205831 +1205832 = 1 In this case the function 119891 119868exp rarr R is called geometric convex function Everygeometric convex (usually known as log-convex) function isalso convex (cf [2])
(c) Taking 120572 = 119868 and 120573 = exp one obtains
119891 (1205821119909 + 1205822119910) le 119891 (119909)ln 1205831 119891 (119910)ln 1205832 (1205831 1205832 isin [1 119890] 1205821 1205822 isin [0 1]) (32)
where 12058311205832 = 119890 and 1205821 + 1205822 = 1 and 119891 119868 rarr Rexp is calledanageometric convex function
Case 2 (a) If 120572 = 120573 = 119902119901 in (28) then
119891(((1205821119909)119901 + (1205822119910)119901)1119901)le ((1205821119891 (119909))119901 + (1205822119891 (119910))119901)1119901 (119901 isin R
+) (33)
International Journal of Analysis 7
where 1205821 1205822 isin [0 1] 1205821199011 + 1205821199012 = 1 and 119891 119868119902119901 rarr R119902119901 iscalled 119876119876-convex function
(b) For 120572 = 119902119901 and 120573 = 119868 we write that119891(((1205821119909)119901 + (1205822119910)119901)1119901) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 1205831 1205832 isin [0 1]) (34)
where 1205821199011 + 1205821199012 = 1 1205831 + 1205832 = 1 and 119891 119868119902119901 rarr R is called119876119868-convex function(c) For 120572 = 119868 and 120573 = 119902119901 we obtain that
119891 (1205821119909 + 1205822119910) le ((1205831119891 (119909))119901 + (1205832119891 (119910))119901)1119901 (1205821 1205822 1205831 1205832 isin [0 1])
(35)
where 1205831199011 +1205831199012 = 1 1205821 +1205822 = 1 and 119891 119868 rarr R119902119901 is called 119868119876-convex function
The lowast-convexity of a function 119891 119868120572 rarr R120573 means geo-metrically that the lowast-points of the graph of 119891 are under thechord joining the endpoints (119886 119891(119886)) and (119887 119891(119887)) on non-Newtonian coordinate system for every 119886 119887 isin 119868120572 By takinginto account the definition of lowast-slope in Definition 3 we have
(119891 (119909) minus 119891 (119886)) 120580 (119909 minus 119886)le (119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886) (36)
which implies
119891 (119909) le 119891 (119886) + ((119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886))times 120580 (119909 minus 119886) (37)
for all 119909 isin [ ]On the other hand (37) means that if 119875119876 and 119877 are any
three lowast-points on the graph of 119891 with 119876 between 119875 and 119877then 119876 is on or below chord 119875119877 In terms of lowast-slope it isequivalent to
119898lowast (119875119876) le119898lowast (119875119877) le119898lowast (119876119877) (38)
with strict inequalities when 119891 is strictly lowast-convexNow to avoid the repetition of the similar statements we
give some necessary theorems and lemmas
Lemma 14 (Jensenrsquos inequality) A 120573-real-valued function 119891defined on an interval 119868120572 is lowast-convex if and only if
119891( 119899
120572sum119896=1
120582119896 times 119909119896) le 119899
120573sum119896=1
120583119896 times 119891 (119909119896) (39)
holds where 120572sum119899119896=1 120582119896 = 1 and 120573sum119899119896=1 120583119896 = 1 for all 120582119899 isin [0 1]and 120583119899 isin [0 1]Proof The proof is straightforward hence omitted
Theorem 15 Let 119891 119868120572 rarr R120573 be a lowast-continuous functionThen 119891 is lowast-convex if and only if 119891 is midpoint lowast-convexthat is
1199091 1199092 isin 119868120572 implies
119891 (119860120572 1199091 1199092) le 119860120573 119891 (1199091) 119891 (1199092) (40)
Proof The proof can be easily obtained using the inequality(39) in Lemma 14
Theorem16 (cf [2]) Let 119891 119868exp rarr Rexp be alowast-differentiablefunction (see [19]) on a subinterval 119868exp sube (0infin) Then thefollowing assertions are equivalent
(i) 119891 is bigeometric convex (concave)(ii) The function 119891lowast(119909) is increasing (decreasing)
Corollary 17 A positive 120573-real-valued function 119891 defined onan interval 119868exp is bigeometric convex if and only if
119891 (119909ln12058211 119909ln12058222 119909ln120582119899119899 )le 119891 (1199091)ln1205821 119891 (1199092)ln1205822 119891 (119909119899)ln120582119899
(41)
holds where prod119899119896=1120582119896 = 119890 for all 1199091 1199092 119909119899 isin 119868exp and1205821 1205822 120582119899 isin [1 119890] Besides we have119891 (exp 1199091 1199092 119909119899)le exp 119891 (1199091) 119891 (1199092) 119891 (119909119899) (42)
Corollary 18 A 120573-real-valued function119891 defined on an inter-val 119868119902119901 is 119876119876-convex if and only if
119891[((12058211199091)119901 + (12058221199092)119901 + sdot sdot sdot + (120582119899119909119899)119901)1119901]le ((1205821119891 (1199091))119901 + (1205822119891 (1199092))119901 + sdot sdot sdot+ (120582119899119891 (119909119899))119901)1119901
(43)
holds where sum119899119896=1 120582119901119896 = 1 for all 1199091 1199092 119909119899 isin 119868119902119901 and1205821 1205822 120582119899 isin [0 1] Thus we have
119891 (119901 1199091 1199092 119909119899)le 119901 119891 (1199091) 119891 (1199092) 119891 (119909119899) (119901 isin R
+) (44)
5 An Application of Multiplicative Continuity
In this section based on the definition of bigeometric convexfunction and multiplicative continuity we get an analogue ofordinary Lipschitz condition on any closed interval
Let 119891 be a bigeometric (multiplicative) convex functionand finite on a closed interval [119909 119910] sub R+ It is obvious that119891is bounded from above by 119872 = max119891(119909) 119891(119910) sincefor any 119911 = 1199091205821199101minus120582 in the interval 119891(119911) le 119891(119909)120582119891(119910)1minus120582 for
8 International Journal of Analysis
120582 isin [1 119890] It is also bounded from below as we see by writingan arbitrary point in the form 119905radic119909119910 for 119905 isin R+ Then
1198912 (radic119909119910) le 119891 (119905radic119909119910)119891(radic119909119910119905 ) (45)
Using 119872 as the upper bound 119891(radic119909119910119905) we obtain119891 (119905radic119909119910) ge 11198721198912 (radic119909119910) = 119898 (46)
Thus a bigeometric convex functionmay not be continuous atthe boundary points of its domainWewill prove that for anyclosed subinterval [119909 119910] of the interior of the domain thereis a constant 119870 gt 0 so that for any two points 119886 119887 isin [119909 119910] subR+
119891 (119886)119891 (119887) le (119886119887)119870 (47)
A function that satisfies (47) for some 119870 and all 119886 and 119887 inan interval is said to satisfy bigeometric Lipschitz conditionon the interval
Theorem 19 Suppose that 119891 119868 rarr R+ is multiplicative con-vex Then 119891 satisfies the multiplicative Lipschitz condition onany closed interval [119909 119910] sub R+ contained in the interior 1198680 of119868 that is 119891 is continuous on 1198680Proof Take 120576 gt 1 so that [119909120576 119910120576] isin 119868 and let 119898 and 119872 bethe lower and upper bounds for 119891 on [119909120576 119910120576] If 119903 and 119904 aredistinct points of [119909 119910] with 119904 gt 119903 set
119911 = 119904120576120582 = ( 119904119903)
1 ln(120576119904119903) (120582 isin (1 119890)) (48)
Then 119911 isin [119909120576 119910120576] and 119904 = 119911ln1205821199031minusln120582 and we obtain
119891 (119904) le 119891 (119911)ln120582 119891 (119903)1minusln120582 = (119891 (119911)119891 (119903) )ln120582 119891 (119903) (49)
which yields
ln(119891 (119904)119891 (119903)) le ln 120582 ln(119891 (119911)119891 (119903) )lt ln( 119904119903)
1 ln(120576)ln(119872119898 )
119891 (119904)119891 (119903) le ( 119904119903)ln(119872119898) ln(120576)
(50)
where 119870 = ln(119872119898) ln(120576) gt 0 Since the points 119903 119904 isin [119909 119910]are arbitrary we get 119891 that satisfies a multiplicative Lipschitzcondition The remaining part can be obtained in the similarway by taking 119904 lt 119903 and 119911 = 119904120576 Finally 119891 is continuoussince [119909 119910] is arbitrary in 1198680
6 Concluding Remarks
Although all arithmetics are isomorphic only by distinguish-ing among them do we obtain suitable tools for construct-ing all the non-Newtonian calculi But the usefulness ofarithmetic is not limited to the construction of calculi webelieve there is a more fundamental reason for consideringalternative arithmetics theymay also be helpful in developingand understanding new systems of measurement that couldyield simpler physical laws
In this paper it was shown that due to the choice ofgenerator function 119860119901 119866119901 and 119867119901 means are reduced toordinary arithmetic geometric and harmonic mean respec-tively As shown in Table 1 for increasing values of 119901 119860119901and 119901 means increase especially 119901 rarr infin these meansconverge to the value of max119909119899 Conversely for increasingvalues of 119901 119867119901 and 119901 means decrease especially 119901 rarr infinthese means converge to the value of min119909119899 Additionallywe give some new definitions regarding convex functionswhich are plotted on the non-Newtonian coordinate systemObviously for different generator functions one can obtainsome new geometrical interpretations of convex functionsOur future works will include the most famous HermiteHadamard inequality for the class of lowast-convex functionsCompeting Interests
The authors declare that they have no competing interests
References
[1] C Niculescu and L-E Persson Convex Functions and TheirApplications A Contemporary Approach Springer Berlin Ger-many 2006
[2] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000
[3] R Webster Convexity Oxford University Press New York NYUSA 1995
[4] J Banas and A Ben Amar ldquoMeasures of noncompactness inlocally convex spaces and fixed point theory for the sum oftwo operators on unbounded convex setsrdquo CommentationesMathematicae Universitatis Carolinae vol 54 no 1 pp 21ndash402013
[5] J Matkowski ldquoGeneralized weighted and quasi-arithmeticmeansrdquo Aequationes Mathematicae vol 79 no 3 pp 203ndash2122010
[6] D Głazowska and J Matkowski ldquoAn invariance of geometricmean with respect to Lagrangian meansrdquo Journal of Mathemat-ical Analysis and Applications vol 331 no 2 pp 1187ndash1199 2007
[7] J Matkowski ldquoGeneralized weighted quasi-arithmetic meansand the Kolmogorov-Nagumo theoremrdquo Colloquium Mathe-maticum vol 133 no 1 pp 35ndash49 2013
[8] N Merentes and K Nikodem ldquoRemarks on strongly convexfunctionsrdquo Aequationes Mathematicae vol 80 no 1-2 pp 193ndash199 2010
[9] M Grossman Bigeometric Calculus Archimedes FoundationBox 240 Rockport Mass USA 1983
[10] M Grossman and R Katz Non-Newtonian Calculus NewtonInstitute 1978
International Journal of Analysis 9
[11] M Grossman The First Nonlinear System of Differential andIntegral Calculus 1979
[12] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008
[13] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007
[14] E Mısırlı and Y Gurefe ldquoMultiplicative adams-bashforth-moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011
[15] A F Cakmak and F Basar ldquoSome new results on sequencespaces with respect to non-Newtonian calculusrdquo Journal ofInequalities and Applications vol 2012 article 228 2012
[16] A F Cakmak and F Basar ldquoCertain spaces of functions overthe field of non-Newtonian complex numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 236124 12 pages 2014
[17] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013
[18] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014
[19] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015
[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014
[21] U Kadak ldquoNon-Newtonian fuzzy numbers and related applica-tionsrdquo Iranian Journal of Fuzzy Systems vol 12 no 5 pp 117ndash1372015
[22] Y Gurefe U Kadak E Misirli and A Kurdi ldquoA new look atthe classical sequence spaces by using multiplicative calculusrdquoUniversity Politehnica of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 78 no 2 pp 9ndash20 2016
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Stochastic AnalysisInternational Journal of
International Journal of Analysis 3
and 120572radic119909 = 119909(12)120572 = 119910 provided there exists an 119910 isin R120572 suchthat 1199102120572 = 11990921 lowast-Arithmetic Suppose that 120572 and 120573 are two arbitrarilyselected generators and (ldquostar-rdquo) also is the ordered pairof arithmetics (120573-arithmetic and 120572-arithmetic) The sets(R120573 + minus times lt) and (R120572 + minus times lt) are complete orderedfields and 119887119890119905119886(119886119897119901ℎ119886)-generator generates 119887119890119905119886(119886119897119901ℎ119886)-arithmetic respectively Definitions given for 120573-arithmeticare also valid for 120572-arithmetic Also 120572-arithmetic is used forarguments and 120573-arithmetic is used for values in particu-lar changes in arguments and values are measured by 120572-differences and 120573-differences respectively
Let 119909 isin (R120572 + minus times lt) and 119910 isin (R120573 + minus times lt) bearbitrarily chosen elements from corresponding arithmeticThen the ordered pair (119909 119910) is called a lowast-point and the setof all lowast-points is called the set of lowast-complex numbers and isdenoted by Clowast that is
Clowast fl 119911lowast = (119909 119910) | 119909 isin R120572 119910 isin R120573 (7)
Definition 2 (see [17]) (a) The lowast-limit of a function 119891 at anelement 119886 inR120572 is if it exists the unique number 119887 inR120573 suchthatlowast lim119909rarr119886
119891 (119909) = 119887 lArrrArrforall120576 gt 0 exist120575 gt 0 ni 1003816100381610038161003816119891 (119909) minus 1198871003816100381610038161003816120573 lt 120576
forall119909 120576 isin R120572 1003816100381610038161003816119909 minus 1198861003816100381610038161003816120572 lt 120575(8)
for 120575 isin R120573 and is written as lowastlim119909rarr119886119891(119909) = 119887A function119891 islowast-continuous at a point 119886 inR120572 if and only
if 119886 is an argument of 119891 and lowastlim119909rarr119886119891(119909) = 119891(119886) When120572 and 120573 are the identity function 119868 the concepts of lowast-limitand lowast-continuity are reduced to those of classical limit andclassical continuity
(b) The isomorphism from 120572-arithmetic to 120573-arithmeticis the unique function 120580 (iota) which has the following threeproperties
(i) 120580 is one to one(ii) 120580 is from R120572 to R120573
(iii) For any numbers 119906 V isin R120572
120580 (119906 + V) = 120580 (119906) + 120580 (V) 120580 (119906 minus V) = 120580 (119906) minus 120580 (V) 120580 (119906 times V) = 120580 (119906) times 120580 (V) 120580 (119906 V) = 120580 (119906) 120580 (V)
(9)
It turns out that 120580(119909) = 120573120572minus1(119909) for every 119909 in R120572 andthat 120580() = for every 120572-integer Since for example119906 + V = 120580minus1120580(119906) + 120580(V) it should be clear that any statement in
120572-arithmetic can readily be transformed into a statement in120573-arithmetic
Definition 3 (see [10]) The following statements are valid
(i) The lowast-points 1198751 1198752 and 1198753 are lowast-collinear providedthat at least one of the following holds
119889lowast (1198752 1198751) + 119889lowast (1198751 1198753) = 119889lowast (1198752 1198753) 119889lowast (1198751 1198752) + 119889lowast (1198752 1198753) = 119889lowast (1198751 1198753) 119889lowast (1198751 1198753) + 119889lowast (1198753 1198752) = 119889lowast (1198751 1198752)
(10)
(ii) A lowast-line is a set 119871 of at least two distinct points suchthat for any distinct points1198751 and1198752 in 119871 a point1198753 isin 119871 if and only if 1198751 1198752 and 1198753 are lowast-collinear When120572 = 120573 = 119868 the lowast-lines are the straight lines in two-dimensional Euclidean space
(iii) The lowast-slope of a lowast-line through the points (1198861 1198871) and(1198862 1198872) is given by
119898lowast = (1198872 minus 1198871) 120580 (1198862 minus 1198861)= 120573 120573minus1 (1198872) minus 120573minus1 (1198871)120572minus1 (1198862) minus 120572minus1 (1198861) (1198861 = 1198862) (11)
for 1198861 1198862 isin R120572 and 1198871 1198872 isin R120573
If the following lowast-limit in (12) exists we denote it by119891lowast(119905) call it the lowast-derivative of 119891 at 119905 and say that 119891 is lowast-differentiable at 119905 (see [19])
lowastlim119909rarr119905
(119891 (119909) minus 119891 (119905)) 120580 (119909 minus 119905)= lim119909rarr119905
120573120573minus1 119891 (119909) minus 120573minus1 119891 (119905)120572minus1 (119909) minus 120572minus1 (119905)
= lim119909rarr119905
120573120573minus1 119891 (119909) minus 120573minus1 119891 (119905)119909 minus 119905sdot 119909 minus 119905120572minus1 (119909) minus 120572minus1 (119905) = 120573
(120573minus1 ∘ 119891)1015840 (119905)(120572minus1)1015840 (119905)
(12)
3 Non-Newtonian (Weighted) Means
Definition 4 (120572-arithmetic mean) Consider that 119899 positivereal numbers 1199091 1199092 119909119899 are given The 120572-mean (average)
4 International Journal of Analysis
denoted by 119860120572 is the 120572-sum of 119909119899rsquos 120572-divided by for all119899 isin N That is
119860120572 = 119899
120572sum119896=1
119909119896 = 119899
120572sum119896=1
120572120572minus1 (119909119896)119899
= 120572120572minus1 (1199091) + 120572minus1 (1199092) + sdot sdot sdot + 120572minus1 (119909119899)119899 (13)
For 120572 = exp we obtain that
119860exp = ( 119899prod119896=1
119909119896)1119899
= (1199091 sdot 1199092 sdot sdot sdot 119909119899)1119899 (14)
Similarly for 120572 = 119902119901 we get
119860119901 = (1199091199011 + 1199091199012 + sdot sdot sdot + 119909119901119899119899 )1119901 119901 isin R 0 (15)
119860exp and 119860119901 are called multiplicative arithmetic mean and119901-arithmetic mean (as usually known p-mean) respectivelyOne can conclude that 119860119901 reduces to arithmetic mean andharmonic mean in the ordinary sense for 119901 = 1 and 119901 = minus1respectively
Remark 5 It is clear that Definition 4 can be written byusing various generators In particular if we take120573-arithmeticinstead of 120572-arithmetic then the mean can be defined by
119860120573 = 119899
120573sum119896=1
119909119896 (16)
Definition 6 (120572-geometric mean) Let 1199091 1199092 119909119899 isin R+The 120572-geometric mean namely 119866120572 is 119899th 120572-root of the 120572-product of (119909119899)rsquos
119866120572 = ( 119899
120572prod119896=1
119909119896)(1119899)120572 = 120572
( 119899prod119896=1
120572minus1 (119909119896))1119899
= 120572(120572minus1 (1199091) 120572minus1 (1199092) 120572minus1 (119909119899))1119899 (17)
We conclude similarly by taking the generators 120572 = expor 120572 = 119902119901 that the 120572-geometric mean can be interpreted asfollows
119866exp = exp (ln1199091 ln1199092 ln119909119899)1119899 (119909119899 gt 1)119866119901 = (1199091199011 1199091199012 119909119901119899)11198991119901 = (1199091 1199092 119909119899)1119899
(119901 = 0) (18)
119866exp and 119866119901 are called multiplicative geometric mean and119901-geometric mean respectively It would clearly have 119866119901 =119860exp for 119901 = 1Definition 7 (120572-harmonic mean) Let 1199091 1199092 119909119899 isin R+ and120572minus1(119909119899) = 0 for each 119899 isin N The 120572-harmonic mean 119867120572 isdefined by
119867120572 = 119899120572sum119896=1
(1 119909119896)= (1 1199091 + 1 1199092 + sdot sdot sdot + 1 119909119899)= 120572 1198991120572minus1 (1199091) + 1120572minus1 (1199092) + sdot sdot sdot + 1120572minus1 (119909119899)
(19)
Similarly one obtains that
119867exp
= exp 1198991ln (1199091) + 1ln (1199092) + sdot sdot sdot + 1ln (119909119899) (119909119899 gt 1)
119867119901 = 1198991 (1199091)119901 + 1 (1199092)119901 + sdot sdot sdot + 1 (119909119899)119901
1119901
(20)
119867exp and119867119901 are called multiplicative harmonic mean and 119901-harmonic mean respectively Obviously the inclusion (20) isreduced to ordinary harmonic mean and ordinary arithmeticmean for 119901 = 1 and 119901 = minus1 respectively
31 Non-Newtonian Weighted Means The weighted mean issimilar to an arithmetic mean where instead of each of thedata points contributing equally to the final average somedata points contributemore than othersMoreover the notionof weighted mean plays a role in descriptive statistics andalso occurs in a more general form in several other areas ofmathematics
The following definitions can give the relationshipsbetween the non-Newtonian weighted means and ordinaryweighted means
Definition 8 (weighted 120572-arithmetic mean) Formally theweighted 120572-arithmetic mean of a nonempty set of data1199091 1199092 119909119899 with nonnegative weights 1199081 1199082 119908119899 isthe quantity
International Journal of Analysis 5
120572 = 119899
120572sum119894=1
(119909119894 times 119894) 119899120572sum119894=1
119894 = (1199091 times 1 + 1199092 times 2 + sdot sdot sdot + 119909119899 times 119899) (1 + 2 + sdot sdot sdot + 119899) = 120572sum119899119894=1 119908119894120572minus1 (119909119894)sum119899119894=1 119908119894 (21)
The formulas are simplified when the weights are 120572-normalized such that they 120572-sum up to 120572sum119899119894=1119894 = 1 Forsuch normalized weights the weighted 120572-arithmetic mean issimply 120572 = 120572sum119899119894=1 119909119894 times 119894 Note that if all the weights areequal the weighted 120572-arithmetic mean is the same as the 120572-arithmetic mean
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-arithmeticmean can be given with the weights 1199081 1199082 119908119899 as fol-lows
exp = expsum119899119894=1 ln (119909119894) 119908119894sum119899119894=1 119908119894 = ( 119899prod119894=1
119909119908119894119894 )1sum119899119894=1 119908119894 (119909119899 gt 1)
119902119901 = 1199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)1199011199081 + 1199082 + sdot sdot sdot + 119908119899 1119901
(22)
exp and 119901 are called multiplicative weighted arithmeticmean and weighted 119901-arithmetic mean respectively expturns out to the ordinary weighted geometric mean Alsoone easily can see that 119901 is reduced to ordinary weightedarithmetic mean and weighted harmonic mean for 119901 = 1 and119901 = minus1 respectivelyDefinition 9 (weighted 120572-geometric mean) Given a set ofpositive reals 1199091 1199092 119909119899 and corresponding weights1199081 1199082 119908119899 then the weighted 120572-geometric mean 120572 isdefined by
120572 = 119899
120572prod119894=1
119909119894(119908119894sum
119899119894=1 119908119894)120572 = 119909(1199081(1199081+sdotsdotsdot+119908119899))1205721
times 119909(1199082(1199081+sdotsdotsdot+119908119899))1205722 times sdot sdot sdot times 119909(119908119899(1199081+sdotsdotsdot+119908119899))120572119899
= 120572[120572minus1 (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [120572minus1 (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [120572minus1 (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(23)
Note that if all the weights are equal the weighted 120572-geometric mean is the same as the 120572-geometric meanTaking 120572 = exp and 120572 = 119902119901 theweighted 120572-geometricmeancan be written for the weights 1199081 1199082 119908119899 as follows
exp = exp [ln (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [ln (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [ln (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(119909119899 gt 1) 119901 = 1199091199011199081(1199081+sdotsdotsdot+119908119899)1 1199091199011199082(1199081+sdotsdotsdot+119908119899)2 119909119901119908119899(1199081+sdotsdotsdot+119908119899)119899 1119901 = ( 119899prod
119894=1
119909119908119894119894 )1sum119899119894=1 119908119894
(24)
exp and 119901 are called weighted multiplicative geometricmean and weighted 119902-geometric mean Also we have 119901 =exp for all 119909119899 gt 1Definition 10 (weighted 120572-harmonic mean) If a set 11990811199082 119908119899 of weights is associated with the data set1199091 1199092 119909119899 then the weighted 120572-harmonic mean isdefined by
120572 = 119899
120572sum119896=1
119896 119899120572sum119896=1
(119896 119909119896)= (1 + sdot sdot sdot + 119899) (1 1199091 + 2 1199092 + sdot sdot sdot + 119899 119909119899)= 120572 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081120572minus1 (1199091) + 1199082120572minus1 (1199092) + sdot sdot sdot + 119908119899120572minus1 (119909119899)
(25)
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-harmonicmean with the weights 1199081 1199082 119908119899 can be written asfollows
exp
= exp 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 ln1199091 + 1199082 ln1199092 + sdot sdot sdot + 119908119899 ln119909119899 (119909119899 gt 1)
119901= 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)119901
1119901
(26)
exp and 119901 are called multiplicative weighted harmonicmean and weighted 119901-geometric mean respectively It isobvious that 119901 is reduced to ordinary weighted harmonicmean and ordinary weighted arithmetic mean for 119901 = 1 and119901 = minus1 respectively
6 International Journal of Analysis
Table 1 Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means
Weight Data 119901 119860119901 119866119901 119867119901 119901 119901 119901 119866exp 119867exp exp exp1199081 = 2 1199091 = 15 10 2750 2474 2264 3282 2988 2727 2402 2336 2905 28261199082 = 5 1199092 = 20 01 2499 2474 2450 3016 2988 2959 2402 2336 2905 28261199083 = 7 1199093 = 25 20 3061 2474 2114 3577 2988 2521 2402 2336 2905 28261199084 = 9 1199094 = 50 50 3822 2474 1873 4169 2988 2155 2402 2336 2905 2826
In Table 1 the non-Newtonian means are obtained byusing different generating functions For 120572 = 119902119901 the119901-means119860119901 119866119901 and119867119901 are reduced to ordinary arithmeticmean geo-metric mean and harmonic mean respectively In particularsome changes are observed for each value of 119860119901 119866119901 and119867119901means depending on the choice of 119901 As shown in the tablefor increasing values of 119901 the 119901-arithmetic mean 119860119901 and itsweighted form 119901 increase in particular 119901 tends to infin andthese means converge to the value of max119909119899 Converselyfor increasing values of 119901 the 119901-harmonic mean 119867119901 andits weighted forms 119901 decrease In particular these meansconverge to the value of min119909119899 as 119901 rarr infin Dependingon the choice of 119901 weighted forms 119901 and 119901 can beincreased or decreased without changing any weights Forthis reason this approach brings a new perspective to theconcept of classical (weighted) mean Moreover when wecompare 119867exp and ordinary harmonic mean in Table 1 wealso see that ordinary harmonic mean is smaller than 119867expOn the contrary119860exp and119866exp are smaller than their classicalforms 119860119901 and 119866119901 for 119901 = 1 Therefore we assert thatthe values of 119866exp 119867exp exp and exp should be evaluatedsatisfactorily
Corollary 11 Consider 119899 positive real numbers 1199091 1199092 119909119899Then the conditions119867120572 lt 119866120572 lt 119860120572 and 120572 lt 120572 lt 120572 holdwhen 120572 = exp for all 119909119899 gt 1 and 120572 = 119902119901 for all 119901 isin R+
4 Non-Newtonian Convexity
In this section the notion of non-Newtonian convex (lowast-convex) functions will be given by using different genera-tors Furthermore the relationships between lowast-convexity andnon-Newtonian weighted mean will be determined
Definition 12 (generalized lowast-convex function) Let 119868120572 be aninterval in R120572 Then 119891 119868120572 rarr R120573 is said to be lowast-convex if
119891 (1205821 times 119909 + 1205822 times 119910) le 1205831 times 119891 (119909) + 1205832 times 119891 (119910) (27)
holds where 1205821 + 1205822 = 1 and 1205831 + 1205832 = 1 for all 1205821 1205822 isin[0 1] and 1205831 1205832 isin [0 1] Therefore by combining this withthe generators 120572 and 120573 we deduce that
119891 (120572 120572minus1 (1205821) 120572minus1 (119909) + 120572minus1 (1205822) 120572minus1 (119910))le 120573 120573minus1 (1205831) 120573minus1119891 (119909) + 120573minus1 (1205832) 120573minus1119891 (119910) (28)
If (28) is strict for all 119909 = 119910 then 119891 is said to be strictly lowast-convex If the inequality in (28) is reversed then 119891 is said to
be lowast-concave On the other hand the inclusion (28) can bewritten with respect to the weighted 120572-arithmetic mean in(21) as follows
119891 (120572 119909 119910) le 120573 119891 (119909) 119891 (119910) (29)
Remark 13 We remark that the definition of lowast-convexity in(27) can be evaluated by non-Newtonian coordinate systeminvolving lowast-lines (see Definition 3) For 120572 = 120573 = 119868 the lowast-lines are straight lines in two-dimensional Euclidean spaceFor this reason we say that almost all the properties ofordinary Cartesian coordinate system will be valid for non-Newtonian coordinate system under lowast-arithmetic
Also depending on the choice of generator functionsthe definition of lowast-convexity in (27) can be interpreted asfollows
Case 1 (a) If we take 120572 = 120573 = exp and 1205821 = 1205831 1205822 = 1205832 in(28) then
119891 (119909ln1205821119910ln1205822) le 119891 (119909)ln1205821 119891 (119910)ln1205822 (1205821 1205822 isin [1 119890])
(30)
where 12058211205822 = 119890 holds and 119891 119868exp rarr Rexp = (0infin) iscalled bigeometric (usually known as multiplicative) convexfunction (cf [2]) Equivalently 119891 is bigeometric convex ifand only if log119891(119909) is an ordinary convex function
(b) For 120572 = exp and 120573 = 119868 we have119891 (119909ln1205821119910ln1205822) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 isin [1 119890] 1205831 1205832 isin [0 1]) (31)
where 12058211205822 = 119890 and 1205831 +1205832 = 1 In this case the function 119891 119868exp rarr R is called geometric convex function Everygeometric convex (usually known as log-convex) function isalso convex (cf [2])
(c) Taking 120572 = 119868 and 120573 = exp one obtains
119891 (1205821119909 + 1205822119910) le 119891 (119909)ln 1205831 119891 (119910)ln 1205832 (1205831 1205832 isin [1 119890] 1205821 1205822 isin [0 1]) (32)
where 12058311205832 = 119890 and 1205821 + 1205822 = 1 and 119891 119868 rarr Rexp is calledanageometric convex function
Case 2 (a) If 120572 = 120573 = 119902119901 in (28) then
119891(((1205821119909)119901 + (1205822119910)119901)1119901)le ((1205821119891 (119909))119901 + (1205822119891 (119910))119901)1119901 (119901 isin R
+) (33)
International Journal of Analysis 7
where 1205821 1205822 isin [0 1] 1205821199011 + 1205821199012 = 1 and 119891 119868119902119901 rarr R119902119901 iscalled 119876119876-convex function
(b) For 120572 = 119902119901 and 120573 = 119868 we write that119891(((1205821119909)119901 + (1205822119910)119901)1119901) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 1205831 1205832 isin [0 1]) (34)
where 1205821199011 + 1205821199012 = 1 1205831 + 1205832 = 1 and 119891 119868119902119901 rarr R is called119876119868-convex function(c) For 120572 = 119868 and 120573 = 119902119901 we obtain that
119891 (1205821119909 + 1205822119910) le ((1205831119891 (119909))119901 + (1205832119891 (119910))119901)1119901 (1205821 1205822 1205831 1205832 isin [0 1])
(35)
where 1205831199011 +1205831199012 = 1 1205821 +1205822 = 1 and 119891 119868 rarr R119902119901 is called 119868119876-convex function
The lowast-convexity of a function 119891 119868120572 rarr R120573 means geo-metrically that the lowast-points of the graph of 119891 are under thechord joining the endpoints (119886 119891(119886)) and (119887 119891(119887)) on non-Newtonian coordinate system for every 119886 119887 isin 119868120572 By takinginto account the definition of lowast-slope in Definition 3 we have
(119891 (119909) minus 119891 (119886)) 120580 (119909 minus 119886)le (119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886) (36)
which implies
119891 (119909) le 119891 (119886) + ((119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886))times 120580 (119909 minus 119886) (37)
for all 119909 isin [ ]On the other hand (37) means that if 119875119876 and 119877 are any
three lowast-points on the graph of 119891 with 119876 between 119875 and 119877then 119876 is on or below chord 119875119877 In terms of lowast-slope it isequivalent to
119898lowast (119875119876) le119898lowast (119875119877) le119898lowast (119876119877) (38)
with strict inequalities when 119891 is strictly lowast-convexNow to avoid the repetition of the similar statements we
give some necessary theorems and lemmas
Lemma 14 (Jensenrsquos inequality) A 120573-real-valued function 119891defined on an interval 119868120572 is lowast-convex if and only if
119891( 119899
120572sum119896=1
120582119896 times 119909119896) le 119899
120573sum119896=1
120583119896 times 119891 (119909119896) (39)
holds where 120572sum119899119896=1 120582119896 = 1 and 120573sum119899119896=1 120583119896 = 1 for all 120582119899 isin [0 1]and 120583119899 isin [0 1]Proof The proof is straightforward hence omitted
Theorem 15 Let 119891 119868120572 rarr R120573 be a lowast-continuous functionThen 119891 is lowast-convex if and only if 119891 is midpoint lowast-convexthat is
1199091 1199092 isin 119868120572 implies
119891 (119860120572 1199091 1199092) le 119860120573 119891 (1199091) 119891 (1199092) (40)
Proof The proof can be easily obtained using the inequality(39) in Lemma 14
Theorem16 (cf [2]) Let 119891 119868exp rarr Rexp be alowast-differentiablefunction (see [19]) on a subinterval 119868exp sube (0infin) Then thefollowing assertions are equivalent
(i) 119891 is bigeometric convex (concave)(ii) The function 119891lowast(119909) is increasing (decreasing)
Corollary 17 A positive 120573-real-valued function 119891 defined onan interval 119868exp is bigeometric convex if and only if
119891 (119909ln12058211 119909ln12058222 119909ln120582119899119899 )le 119891 (1199091)ln1205821 119891 (1199092)ln1205822 119891 (119909119899)ln120582119899
(41)
holds where prod119899119896=1120582119896 = 119890 for all 1199091 1199092 119909119899 isin 119868exp and1205821 1205822 120582119899 isin [1 119890] Besides we have119891 (exp 1199091 1199092 119909119899)le exp 119891 (1199091) 119891 (1199092) 119891 (119909119899) (42)
Corollary 18 A 120573-real-valued function119891 defined on an inter-val 119868119902119901 is 119876119876-convex if and only if
119891[((12058211199091)119901 + (12058221199092)119901 + sdot sdot sdot + (120582119899119909119899)119901)1119901]le ((1205821119891 (1199091))119901 + (1205822119891 (1199092))119901 + sdot sdot sdot+ (120582119899119891 (119909119899))119901)1119901
(43)
holds where sum119899119896=1 120582119901119896 = 1 for all 1199091 1199092 119909119899 isin 119868119902119901 and1205821 1205822 120582119899 isin [0 1] Thus we have
119891 (119901 1199091 1199092 119909119899)le 119901 119891 (1199091) 119891 (1199092) 119891 (119909119899) (119901 isin R
+) (44)
5 An Application of Multiplicative Continuity
In this section based on the definition of bigeometric convexfunction and multiplicative continuity we get an analogue ofordinary Lipschitz condition on any closed interval
Let 119891 be a bigeometric (multiplicative) convex functionand finite on a closed interval [119909 119910] sub R+ It is obvious that119891is bounded from above by 119872 = max119891(119909) 119891(119910) sincefor any 119911 = 1199091205821199101minus120582 in the interval 119891(119911) le 119891(119909)120582119891(119910)1minus120582 for
8 International Journal of Analysis
120582 isin [1 119890] It is also bounded from below as we see by writingan arbitrary point in the form 119905radic119909119910 for 119905 isin R+ Then
1198912 (radic119909119910) le 119891 (119905radic119909119910)119891(radic119909119910119905 ) (45)
Using 119872 as the upper bound 119891(radic119909119910119905) we obtain119891 (119905radic119909119910) ge 11198721198912 (radic119909119910) = 119898 (46)
Thus a bigeometric convex functionmay not be continuous atthe boundary points of its domainWewill prove that for anyclosed subinterval [119909 119910] of the interior of the domain thereis a constant 119870 gt 0 so that for any two points 119886 119887 isin [119909 119910] subR+
119891 (119886)119891 (119887) le (119886119887)119870 (47)
A function that satisfies (47) for some 119870 and all 119886 and 119887 inan interval is said to satisfy bigeometric Lipschitz conditionon the interval
Theorem 19 Suppose that 119891 119868 rarr R+ is multiplicative con-vex Then 119891 satisfies the multiplicative Lipschitz condition onany closed interval [119909 119910] sub R+ contained in the interior 1198680 of119868 that is 119891 is continuous on 1198680Proof Take 120576 gt 1 so that [119909120576 119910120576] isin 119868 and let 119898 and 119872 bethe lower and upper bounds for 119891 on [119909120576 119910120576] If 119903 and 119904 aredistinct points of [119909 119910] with 119904 gt 119903 set
119911 = 119904120576120582 = ( 119904119903)
1 ln(120576119904119903) (120582 isin (1 119890)) (48)
Then 119911 isin [119909120576 119910120576] and 119904 = 119911ln1205821199031minusln120582 and we obtain
119891 (119904) le 119891 (119911)ln120582 119891 (119903)1minusln120582 = (119891 (119911)119891 (119903) )ln120582 119891 (119903) (49)
which yields
ln(119891 (119904)119891 (119903)) le ln 120582 ln(119891 (119911)119891 (119903) )lt ln( 119904119903)
1 ln(120576)ln(119872119898 )
119891 (119904)119891 (119903) le ( 119904119903)ln(119872119898) ln(120576)
(50)
where 119870 = ln(119872119898) ln(120576) gt 0 Since the points 119903 119904 isin [119909 119910]are arbitrary we get 119891 that satisfies a multiplicative Lipschitzcondition The remaining part can be obtained in the similarway by taking 119904 lt 119903 and 119911 = 119904120576 Finally 119891 is continuoussince [119909 119910] is arbitrary in 1198680
6 Concluding Remarks
Although all arithmetics are isomorphic only by distinguish-ing among them do we obtain suitable tools for construct-ing all the non-Newtonian calculi But the usefulness ofarithmetic is not limited to the construction of calculi webelieve there is a more fundamental reason for consideringalternative arithmetics theymay also be helpful in developingand understanding new systems of measurement that couldyield simpler physical laws
In this paper it was shown that due to the choice ofgenerator function 119860119901 119866119901 and 119867119901 means are reduced toordinary arithmetic geometric and harmonic mean respec-tively As shown in Table 1 for increasing values of 119901 119860119901and 119901 means increase especially 119901 rarr infin these meansconverge to the value of max119909119899 Conversely for increasingvalues of 119901 119867119901 and 119901 means decrease especially 119901 rarr infinthese means converge to the value of min119909119899 Additionallywe give some new definitions regarding convex functionswhich are plotted on the non-Newtonian coordinate systemObviously for different generator functions one can obtainsome new geometrical interpretations of convex functionsOur future works will include the most famous HermiteHadamard inequality for the class of lowast-convex functionsCompeting Interests
The authors declare that they have no competing interests
References
[1] C Niculescu and L-E Persson Convex Functions and TheirApplications A Contemporary Approach Springer Berlin Ger-many 2006
[2] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000
[3] R Webster Convexity Oxford University Press New York NYUSA 1995
[4] J Banas and A Ben Amar ldquoMeasures of noncompactness inlocally convex spaces and fixed point theory for the sum oftwo operators on unbounded convex setsrdquo CommentationesMathematicae Universitatis Carolinae vol 54 no 1 pp 21ndash402013
[5] J Matkowski ldquoGeneralized weighted and quasi-arithmeticmeansrdquo Aequationes Mathematicae vol 79 no 3 pp 203ndash2122010
[6] D Głazowska and J Matkowski ldquoAn invariance of geometricmean with respect to Lagrangian meansrdquo Journal of Mathemat-ical Analysis and Applications vol 331 no 2 pp 1187ndash1199 2007
[7] J Matkowski ldquoGeneralized weighted quasi-arithmetic meansand the Kolmogorov-Nagumo theoremrdquo Colloquium Mathe-maticum vol 133 no 1 pp 35ndash49 2013
[8] N Merentes and K Nikodem ldquoRemarks on strongly convexfunctionsrdquo Aequationes Mathematicae vol 80 no 1-2 pp 193ndash199 2010
[9] M Grossman Bigeometric Calculus Archimedes FoundationBox 240 Rockport Mass USA 1983
[10] M Grossman and R Katz Non-Newtonian Calculus NewtonInstitute 1978
International Journal of Analysis 9
[11] M Grossman The First Nonlinear System of Differential andIntegral Calculus 1979
[12] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008
[13] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007
[14] E Mısırlı and Y Gurefe ldquoMultiplicative adams-bashforth-moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011
[15] A F Cakmak and F Basar ldquoSome new results on sequencespaces with respect to non-Newtonian calculusrdquo Journal ofInequalities and Applications vol 2012 article 228 2012
[16] A F Cakmak and F Basar ldquoCertain spaces of functions overthe field of non-Newtonian complex numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 236124 12 pages 2014
[17] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013
[18] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014
[19] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015
[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014
[21] U Kadak ldquoNon-Newtonian fuzzy numbers and related applica-tionsrdquo Iranian Journal of Fuzzy Systems vol 12 no 5 pp 117ndash1372015
[22] Y Gurefe U Kadak E Misirli and A Kurdi ldquoA new look atthe classical sequence spaces by using multiplicative calculusrdquoUniversity Politehnica of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 78 no 2 pp 9ndash20 2016
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Stochastic AnalysisInternational Journal of
4 International Journal of Analysis
denoted by 119860120572 is the 120572-sum of 119909119899rsquos 120572-divided by for all119899 isin N That is
119860120572 = 119899
120572sum119896=1
119909119896 = 119899
120572sum119896=1
120572120572minus1 (119909119896)119899
= 120572120572minus1 (1199091) + 120572minus1 (1199092) + sdot sdot sdot + 120572minus1 (119909119899)119899 (13)
For 120572 = exp we obtain that
119860exp = ( 119899prod119896=1
119909119896)1119899
= (1199091 sdot 1199092 sdot sdot sdot 119909119899)1119899 (14)
Similarly for 120572 = 119902119901 we get
119860119901 = (1199091199011 + 1199091199012 + sdot sdot sdot + 119909119901119899119899 )1119901 119901 isin R 0 (15)
119860exp and 119860119901 are called multiplicative arithmetic mean and119901-arithmetic mean (as usually known p-mean) respectivelyOne can conclude that 119860119901 reduces to arithmetic mean andharmonic mean in the ordinary sense for 119901 = 1 and 119901 = minus1respectively
Remark 5 It is clear that Definition 4 can be written byusing various generators In particular if we take120573-arithmeticinstead of 120572-arithmetic then the mean can be defined by
119860120573 = 119899
120573sum119896=1
119909119896 (16)
Definition 6 (120572-geometric mean) Let 1199091 1199092 119909119899 isin R+The 120572-geometric mean namely 119866120572 is 119899th 120572-root of the 120572-product of (119909119899)rsquos
119866120572 = ( 119899
120572prod119896=1
119909119896)(1119899)120572 = 120572
( 119899prod119896=1
120572minus1 (119909119896))1119899
= 120572(120572minus1 (1199091) 120572minus1 (1199092) 120572minus1 (119909119899))1119899 (17)
We conclude similarly by taking the generators 120572 = expor 120572 = 119902119901 that the 120572-geometric mean can be interpreted asfollows
119866exp = exp (ln1199091 ln1199092 ln119909119899)1119899 (119909119899 gt 1)119866119901 = (1199091199011 1199091199012 119909119901119899)11198991119901 = (1199091 1199092 119909119899)1119899
(119901 = 0) (18)
119866exp and 119866119901 are called multiplicative geometric mean and119901-geometric mean respectively It would clearly have 119866119901 =119860exp for 119901 = 1Definition 7 (120572-harmonic mean) Let 1199091 1199092 119909119899 isin R+ and120572minus1(119909119899) = 0 for each 119899 isin N The 120572-harmonic mean 119867120572 isdefined by
119867120572 = 119899120572sum119896=1
(1 119909119896)= (1 1199091 + 1 1199092 + sdot sdot sdot + 1 119909119899)= 120572 1198991120572minus1 (1199091) + 1120572minus1 (1199092) + sdot sdot sdot + 1120572minus1 (119909119899)
(19)
Similarly one obtains that
119867exp
= exp 1198991ln (1199091) + 1ln (1199092) + sdot sdot sdot + 1ln (119909119899) (119909119899 gt 1)
119867119901 = 1198991 (1199091)119901 + 1 (1199092)119901 + sdot sdot sdot + 1 (119909119899)119901
1119901
(20)
119867exp and119867119901 are called multiplicative harmonic mean and 119901-harmonic mean respectively Obviously the inclusion (20) isreduced to ordinary harmonic mean and ordinary arithmeticmean for 119901 = 1 and 119901 = minus1 respectively
31 Non-Newtonian Weighted Means The weighted mean issimilar to an arithmetic mean where instead of each of thedata points contributing equally to the final average somedata points contributemore than othersMoreover the notionof weighted mean plays a role in descriptive statistics andalso occurs in a more general form in several other areas ofmathematics
The following definitions can give the relationshipsbetween the non-Newtonian weighted means and ordinaryweighted means
Definition 8 (weighted 120572-arithmetic mean) Formally theweighted 120572-arithmetic mean of a nonempty set of data1199091 1199092 119909119899 with nonnegative weights 1199081 1199082 119908119899 isthe quantity
International Journal of Analysis 5
120572 = 119899
120572sum119894=1
(119909119894 times 119894) 119899120572sum119894=1
119894 = (1199091 times 1 + 1199092 times 2 + sdot sdot sdot + 119909119899 times 119899) (1 + 2 + sdot sdot sdot + 119899) = 120572sum119899119894=1 119908119894120572minus1 (119909119894)sum119899119894=1 119908119894 (21)
The formulas are simplified when the weights are 120572-normalized such that they 120572-sum up to 120572sum119899119894=1119894 = 1 Forsuch normalized weights the weighted 120572-arithmetic mean issimply 120572 = 120572sum119899119894=1 119909119894 times 119894 Note that if all the weights areequal the weighted 120572-arithmetic mean is the same as the 120572-arithmetic mean
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-arithmeticmean can be given with the weights 1199081 1199082 119908119899 as fol-lows
exp = expsum119899119894=1 ln (119909119894) 119908119894sum119899119894=1 119908119894 = ( 119899prod119894=1
119909119908119894119894 )1sum119899119894=1 119908119894 (119909119899 gt 1)
119902119901 = 1199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)1199011199081 + 1199082 + sdot sdot sdot + 119908119899 1119901
(22)
exp and 119901 are called multiplicative weighted arithmeticmean and weighted 119901-arithmetic mean respectively expturns out to the ordinary weighted geometric mean Alsoone easily can see that 119901 is reduced to ordinary weightedarithmetic mean and weighted harmonic mean for 119901 = 1 and119901 = minus1 respectivelyDefinition 9 (weighted 120572-geometric mean) Given a set ofpositive reals 1199091 1199092 119909119899 and corresponding weights1199081 1199082 119908119899 then the weighted 120572-geometric mean 120572 isdefined by
120572 = 119899
120572prod119894=1
119909119894(119908119894sum
119899119894=1 119908119894)120572 = 119909(1199081(1199081+sdotsdotsdot+119908119899))1205721
times 119909(1199082(1199081+sdotsdotsdot+119908119899))1205722 times sdot sdot sdot times 119909(119908119899(1199081+sdotsdotsdot+119908119899))120572119899
= 120572[120572minus1 (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [120572minus1 (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [120572minus1 (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(23)
Note that if all the weights are equal the weighted 120572-geometric mean is the same as the 120572-geometric meanTaking 120572 = exp and 120572 = 119902119901 theweighted 120572-geometricmeancan be written for the weights 1199081 1199082 119908119899 as follows
exp = exp [ln (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [ln (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [ln (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(119909119899 gt 1) 119901 = 1199091199011199081(1199081+sdotsdotsdot+119908119899)1 1199091199011199082(1199081+sdotsdotsdot+119908119899)2 119909119901119908119899(1199081+sdotsdotsdot+119908119899)119899 1119901 = ( 119899prod
119894=1
119909119908119894119894 )1sum119899119894=1 119908119894
(24)
exp and 119901 are called weighted multiplicative geometricmean and weighted 119902-geometric mean Also we have 119901 =exp for all 119909119899 gt 1Definition 10 (weighted 120572-harmonic mean) If a set 11990811199082 119908119899 of weights is associated with the data set1199091 1199092 119909119899 then the weighted 120572-harmonic mean isdefined by
120572 = 119899
120572sum119896=1
119896 119899120572sum119896=1
(119896 119909119896)= (1 + sdot sdot sdot + 119899) (1 1199091 + 2 1199092 + sdot sdot sdot + 119899 119909119899)= 120572 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081120572minus1 (1199091) + 1199082120572minus1 (1199092) + sdot sdot sdot + 119908119899120572minus1 (119909119899)
(25)
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-harmonicmean with the weights 1199081 1199082 119908119899 can be written asfollows
exp
= exp 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 ln1199091 + 1199082 ln1199092 + sdot sdot sdot + 119908119899 ln119909119899 (119909119899 gt 1)
119901= 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)119901
1119901
(26)
exp and 119901 are called multiplicative weighted harmonicmean and weighted 119901-geometric mean respectively It isobvious that 119901 is reduced to ordinary weighted harmonicmean and ordinary weighted arithmetic mean for 119901 = 1 and119901 = minus1 respectively
6 International Journal of Analysis
Table 1 Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means
Weight Data 119901 119860119901 119866119901 119867119901 119901 119901 119901 119866exp 119867exp exp exp1199081 = 2 1199091 = 15 10 2750 2474 2264 3282 2988 2727 2402 2336 2905 28261199082 = 5 1199092 = 20 01 2499 2474 2450 3016 2988 2959 2402 2336 2905 28261199083 = 7 1199093 = 25 20 3061 2474 2114 3577 2988 2521 2402 2336 2905 28261199084 = 9 1199094 = 50 50 3822 2474 1873 4169 2988 2155 2402 2336 2905 2826
In Table 1 the non-Newtonian means are obtained byusing different generating functions For 120572 = 119902119901 the119901-means119860119901 119866119901 and119867119901 are reduced to ordinary arithmeticmean geo-metric mean and harmonic mean respectively In particularsome changes are observed for each value of 119860119901 119866119901 and119867119901means depending on the choice of 119901 As shown in the tablefor increasing values of 119901 the 119901-arithmetic mean 119860119901 and itsweighted form 119901 increase in particular 119901 tends to infin andthese means converge to the value of max119909119899 Converselyfor increasing values of 119901 the 119901-harmonic mean 119867119901 andits weighted forms 119901 decrease In particular these meansconverge to the value of min119909119899 as 119901 rarr infin Dependingon the choice of 119901 weighted forms 119901 and 119901 can beincreased or decreased without changing any weights Forthis reason this approach brings a new perspective to theconcept of classical (weighted) mean Moreover when wecompare 119867exp and ordinary harmonic mean in Table 1 wealso see that ordinary harmonic mean is smaller than 119867expOn the contrary119860exp and119866exp are smaller than their classicalforms 119860119901 and 119866119901 for 119901 = 1 Therefore we assert thatthe values of 119866exp 119867exp exp and exp should be evaluatedsatisfactorily
Corollary 11 Consider 119899 positive real numbers 1199091 1199092 119909119899Then the conditions119867120572 lt 119866120572 lt 119860120572 and 120572 lt 120572 lt 120572 holdwhen 120572 = exp for all 119909119899 gt 1 and 120572 = 119902119901 for all 119901 isin R+
4 Non-Newtonian Convexity
In this section the notion of non-Newtonian convex (lowast-convex) functions will be given by using different genera-tors Furthermore the relationships between lowast-convexity andnon-Newtonian weighted mean will be determined
Definition 12 (generalized lowast-convex function) Let 119868120572 be aninterval in R120572 Then 119891 119868120572 rarr R120573 is said to be lowast-convex if
119891 (1205821 times 119909 + 1205822 times 119910) le 1205831 times 119891 (119909) + 1205832 times 119891 (119910) (27)
holds where 1205821 + 1205822 = 1 and 1205831 + 1205832 = 1 for all 1205821 1205822 isin[0 1] and 1205831 1205832 isin [0 1] Therefore by combining this withthe generators 120572 and 120573 we deduce that
119891 (120572 120572minus1 (1205821) 120572minus1 (119909) + 120572minus1 (1205822) 120572minus1 (119910))le 120573 120573minus1 (1205831) 120573minus1119891 (119909) + 120573minus1 (1205832) 120573minus1119891 (119910) (28)
If (28) is strict for all 119909 = 119910 then 119891 is said to be strictly lowast-convex If the inequality in (28) is reversed then 119891 is said to
be lowast-concave On the other hand the inclusion (28) can bewritten with respect to the weighted 120572-arithmetic mean in(21) as follows
119891 (120572 119909 119910) le 120573 119891 (119909) 119891 (119910) (29)
Remark 13 We remark that the definition of lowast-convexity in(27) can be evaluated by non-Newtonian coordinate systeminvolving lowast-lines (see Definition 3) For 120572 = 120573 = 119868 the lowast-lines are straight lines in two-dimensional Euclidean spaceFor this reason we say that almost all the properties ofordinary Cartesian coordinate system will be valid for non-Newtonian coordinate system under lowast-arithmetic
Also depending on the choice of generator functionsthe definition of lowast-convexity in (27) can be interpreted asfollows
Case 1 (a) If we take 120572 = 120573 = exp and 1205821 = 1205831 1205822 = 1205832 in(28) then
119891 (119909ln1205821119910ln1205822) le 119891 (119909)ln1205821 119891 (119910)ln1205822 (1205821 1205822 isin [1 119890])
(30)
where 12058211205822 = 119890 holds and 119891 119868exp rarr Rexp = (0infin) iscalled bigeometric (usually known as multiplicative) convexfunction (cf [2]) Equivalently 119891 is bigeometric convex ifand only if log119891(119909) is an ordinary convex function
(b) For 120572 = exp and 120573 = 119868 we have119891 (119909ln1205821119910ln1205822) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 isin [1 119890] 1205831 1205832 isin [0 1]) (31)
where 12058211205822 = 119890 and 1205831 +1205832 = 1 In this case the function 119891 119868exp rarr R is called geometric convex function Everygeometric convex (usually known as log-convex) function isalso convex (cf [2])
(c) Taking 120572 = 119868 and 120573 = exp one obtains
119891 (1205821119909 + 1205822119910) le 119891 (119909)ln 1205831 119891 (119910)ln 1205832 (1205831 1205832 isin [1 119890] 1205821 1205822 isin [0 1]) (32)
where 12058311205832 = 119890 and 1205821 + 1205822 = 1 and 119891 119868 rarr Rexp is calledanageometric convex function
Case 2 (a) If 120572 = 120573 = 119902119901 in (28) then
119891(((1205821119909)119901 + (1205822119910)119901)1119901)le ((1205821119891 (119909))119901 + (1205822119891 (119910))119901)1119901 (119901 isin R
+) (33)
International Journal of Analysis 7
where 1205821 1205822 isin [0 1] 1205821199011 + 1205821199012 = 1 and 119891 119868119902119901 rarr R119902119901 iscalled 119876119876-convex function
(b) For 120572 = 119902119901 and 120573 = 119868 we write that119891(((1205821119909)119901 + (1205822119910)119901)1119901) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 1205831 1205832 isin [0 1]) (34)
where 1205821199011 + 1205821199012 = 1 1205831 + 1205832 = 1 and 119891 119868119902119901 rarr R is called119876119868-convex function(c) For 120572 = 119868 and 120573 = 119902119901 we obtain that
119891 (1205821119909 + 1205822119910) le ((1205831119891 (119909))119901 + (1205832119891 (119910))119901)1119901 (1205821 1205822 1205831 1205832 isin [0 1])
(35)
where 1205831199011 +1205831199012 = 1 1205821 +1205822 = 1 and 119891 119868 rarr R119902119901 is called 119868119876-convex function
The lowast-convexity of a function 119891 119868120572 rarr R120573 means geo-metrically that the lowast-points of the graph of 119891 are under thechord joining the endpoints (119886 119891(119886)) and (119887 119891(119887)) on non-Newtonian coordinate system for every 119886 119887 isin 119868120572 By takinginto account the definition of lowast-slope in Definition 3 we have
(119891 (119909) minus 119891 (119886)) 120580 (119909 minus 119886)le (119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886) (36)
which implies
119891 (119909) le 119891 (119886) + ((119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886))times 120580 (119909 minus 119886) (37)
for all 119909 isin [ ]On the other hand (37) means that if 119875119876 and 119877 are any
three lowast-points on the graph of 119891 with 119876 between 119875 and 119877then 119876 is on or below chord 119875119877 In terms of lowast-slope it isequivalent to
119898lowast (119875119876) le119898lowast (119875119877) le119898lowast (119876119877) (38)
with strict inequalities when 119891 is strictly lowast-convexNow to avoid the repetition of the similar statements we
give some necessary theorems and lemmas
Lemma 14 (Jensenrsquos inequality) A 120573-real-valued function 119891defined on an interval 119868120572 is lowast-convex if and only if
119891( 119899
120572sum119896=1
120582119896 times 119909119896) le 119899
120573sum119896=1
120583119896 times 119891 (119909119896) (39)
holds where 120572sum119899119896=1 120582119896 = 1 and 120573sum119899119896=1 120583119896 = 1 for all 120582119899 isin [0 1]and 120583119899 isin [0 1]Proof The proof is straightforward hence omitted
Theorem 15 Let 119891 119868120572 rarr R120573 be a lowast-continuous functionThen 119891 is lowast-convex if and only if 119891 is midpoint lowast-convexthat is
1199091 1199092 isin 119868120572 implies
119891 (119860120572 1199091 1199092) le 119860120573 119891 (1199091) 119891 (1199092) (40)
Proof The proof can be easily obtained using the inequality(39) in Lemma 14
Theorem16 (cf [2]) Let 119891 119868exp rarr Rexp be alowast-differentiablefunction (see [19]) on a subinterval 119868exp sube (0infin) Then thefollowing assertions are equivalent
(i) 119891 is bigeometric convex (concave)(ii) The function 119891lowast(119909) is increasing (decreasing)
Corollary 17 A positive 120573-real-valued function 119891 defined onan interval 119868exp is bigeometric convex if and only if
119891 (119909ln12058211 119909ln12058222 119909ln120582119899119899 )le 119891 (1199091)ln1205821 119891 (1199092)ln1205822 119891 (119909119899)ln120582119899
(41)
holds where prod119899119896=1120582119896 = 119890 for all 1199091 1199092 119909119899 isin 119868exp and1205821 1205822 120582119899 isin [1 119890] Besides we have119891 (exp 1199091 1199092 119909119899)le exp 119891 (1199091) 119891 (1199092) 119891 (119909119899) (42)
Corollary 18 A 120573-real-valued function119891 defined on an inter-val 119868119902119901 is 119876119876-convex if and only if
119891[((12058211199091)119901 + (12058221199092)119901 + sdot sdot sdot + (120582119899119909119899)119901)1119901]le ((1205821119891 (1199091))119901 + (1205822119891 (1199092))119901 + sdot sdot sdot+ (120582119899119891 (119909119899))119901)1119901
(43)
holds where sum119899119896=1 120582119901119896 = 1 for all 1199091 1199092 119909119899 isin 119868119902119901 and1205821 1205822 120582119899 isin [0 1] Thus we have
119891 (119901 1199091 1199092 119909119899)le 119901 119891 (1199091) 119891 (1199092) 119891 (119909119899) (119901 isin R
+) (44)
5 An Application of Multiplicative Continuity
In this section based on the definition of bigeometric convexfunction and multiplicative continuity we get an analogue ofordinary Lipschitz condition on any closed interval
Let 119891 be a bigeometric (multiplicative) convex functionand finite on a closed interval [119909 119910] sub R+ It is obvious that119891is bounded from above by 119872 = max119891(119909) 119891(119910) sincefor any 119911 = 1199091205821199101minus120582 in the interval 119891(119911) le 119891(119909)120582119891(119910)1minus120582 for
8 International Journal of Analysis
120582 isin [1 119890] It is also bounded from below as we see by writingan arbitrary point in the form 119905radic119909119910 for 119905 isin R+ Then
1198912 (radic119909119910) le 119891 (119905radic119909119910)119891(radic119909119910119905 ) (45)
Using 119872 as the upper bound 119891(radic119909119910119905) we obtain119891 (119905radic119909119910) ge 11198721198912 (radic119909119910) = 119898 (46)
Thus a bigeometric convex functionmay not be continuous atthe boundary points of its domainWewill prove that for anyclosed subinterval [119909 119910] of the interior of the domain thereis a constant 119870 gt 0 so that for any two points 119886 119887 isin [119909 119910] subR+
119891 (119886)119891 (119887) le (119886119887)119870 (47)
A function that satisfies (47) for some 119870 and all 119886 and 119887 inan interval is said to satisfy bigeometric Lipschitz conditionon the interval
Theorem 19 Suppose that 119891 119868 rarr R+ is multiplicative con-vex Then 119891 satisfies the multiplicative Lipschitz condition onany closed interval [119909 119910] sub R+ contained in the interior 1198680 of119868 that is 119891 is continuous on 1198680Proof Take 120576 gt 1 so that [119909120576 119910120576] isin 119868 and let 119898 and 119872 bethe lower and upper bounds for 119891 on [119909120576 119910120576] If 119903 and 119904 aredistinct points of [119909 119910] with 119904 gt 119903 set
119911 = 119904120576120582 = ( 119904119903)
1 ln(120576119904119903) (120582 isin (1 119890)) (48)
Then 119911 isin [119909120576 119910120576] and 119904 = 119911ln1205821199031minusln120582 and we obtain
119891 (119904) le 119891 (119911)ln120582 119891 (119903)1minusln120582 = (119891 (119911)119891 (119903) )ln120582 119891 (119903) (49)
which yields
ln(119891 (119904)119891 (119903)) le ln 120582 ln(119891 (119911)119891 (119903) )lt ln( 119904119903)
1 ln(120576)ln(119872119898 )
119891 (119904)119891 (119903) le ( 119904119903)ln(119872119898) ln(120576)
(50)
where 119870 = ln(119872119898) ln(120576) gt 0 Since the points 119903 119904 isin [119909 119910]are arbitrary we get 119891 that satisfies a multiplicative Lipschitzcondition The remaining part can be obtained in the similarway by taking 119904 lt 119903 and 119911 = 119904120576 Finally 119891 is continuoussince [119909 119910] is arbitrary in 1198680
6 Concluding Remarks
Although all arithmetics are isomorphic only by distinguish-ing among them do we obtain suitable tools for construct-ing all the non-Newtonian calculi But the usefulness ofarithmetic is not limited to the construction of calculi webelieve there is a more fundamental reason for consideringalternative arithmetics theymay also be helpful in developingand understanding new systems of measurement that couldyield simpler physical laws
In this paper it was shown that due to the choice ofgenerator function 119860119901 119866119901 and 119867119901 means are reduced toordinary arithmetic geometric and harmonic mean respec-tively As shown in Table 1 for increasing values of 119901 119860119901and 119901 means increase especially 119901 rarr infin these meansconverge to the value of max119909119899 Conversely for increasingvalues of 119901 119867119901 and 119901 means decrease especially 119901 rarr infinthese means converge to the value of min119909119899 Additionallywe give some new definitions regarding convex functionswhich are plotted on the non-Newtonian coordinate systemObviously for different generator functions one can obtainsome new geometrical interpretations of convex functionsOur future works will include the most famous HermiteHadamard inequality for the class of lowast-convex functionsCompeting Interests
The authors declare that they have no competing interests
References
[1] C Niculescu and L-E Persson Convex Functions and TheirApplications A Contemporary Approach Springer Berlin Ger-many 2006
[2] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000
[3] R Webster Convexity Oxford University Press New York NYUSA 1995
[4] J Banas and A Ben Amar ldquoMeasures of noncompactness inlocally convex spaces and fixed point theory for the sum oftwo operators on unbounded convex setsrdquo CommentationesMathematicae Universitatis Carolinae vol 54 no 1 pp 21ndash402013
[5] J Matkowski ldquoGeneralized weighted and quasi-arithmeticmeansrdquo Aequationes Mathematicae vol 79 no 3 pp 203ndash2122010
[6] D Głazowska and J Matkowski ldquoAn invariance of geometricmean with respect to Lagrangian meansrdquo Journal of Mathemat-ical Analysis and Applications vol 331 no 2 pp 1187ndash1199 2007
[7] J Matkowski ldquoGeneralized weighted quasi-arithmetic meansand the Kolmogorov-Nagumo theoremrdquo Colloquium Mathe-maticum vol 133 no 1 pp 35ndash49 2013
[8] N Merentes and K Nikodem ldquoRemarks on strongly convexfunctionsrdquo Aequationes Mathematicae vol 80 no 1-2 pp 193ndash199 2010
[9] M Grossman Bigeometric Calculus Archimedes FoundationBox 240 Rockport Mass USA 1983
[10] M Grossman and R Katz Non-Newtonian Calculus NewtonInstitute 1978
International Journal of Analysis 9
[11] M Grossman The First Nonlinear System of Differential andIntegral Calculus 1979
[12] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008
[13] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007
[14] E Mısırlı and Y Gurefe ldquoMultiplicative adams-bashforth-moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011
[15] A F Cakmak and F Basar ldquoSome new results on sequencespaces with respect to non-Newtonian calculusrdquo Journal ofInequalities and Applications vol 2012 article 228 2012
[16] A F Cakmak and F Basar ldquoCertain spaces of functions overthe field of non-Newtonian complex numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 236124 12 pages 2014
[17] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013
[18] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014
[19] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015
[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014
[21] U Kadak ldquoNon-Newtonian fuzzy numbers and related applica-tionsrdquo Iranian Journal of Fuzzy Systems vol 12 no 5 pp 117ndash1372015
[22] Y Gurefe U Kadak E Misirli and A Kurdi ldquoA new look atthe classical sequence spaces by using multiplicative calculusrdquoUniversity Politehnica of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 78 no 2 pp 9ndash20 2016
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Stochastic AnalysisInternational Journal of
International Journal of Analysis 5
120572 = 119899
120572sum119894=1
(119909119894 times 119894) 119899120572sum119894=1
119894 = (1199091 times 1 + 1199092 times 2 + sdot sdot sdot + 119909119899 times 119899) (1 + 2 + sdot sdot sdot + 119899) = 120572sum119899119894=1 119908119894120572minus1 (119909119894)sum119899119894=1 119908119894 (21)
The formulas are simplified when the weights are 120572-normalized such that they 120572-sum up to 120572sum119899119894=1119894 = 1 Forsuch normalized weights the weighted 120572-arithmetic mean issimply 120572 = 120572sum119899119894=1 119909119894 times 119894 Note that if all the weights areequal the weighted 120572-arithmetic mean is the same as the 120572-arithmetic mean
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-arithmeticmean can be given with the weights 1199081 1199082 119908119899 as fol-lows
exp = expsum119899119894=1 ln (119909119894) 119908119894sum119899119894=1 119908119894 = ( 119899prod119894=1
119909119908119894119894 )1sum119899119894=1 119908119894 (119909119899 gt 1)
119902119901 = 1199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)1199011199081 + 1199082 + sdot sdot sdot + 119908119899 1119901
(22)
exp and 119901 are called multiplicative weighted arithmeticmean and weighted 119901-arithmetic mean respectively expturns out to the ordinary weighted geometric mean Alsoone easily can see that 119901 is reduced to ordinary weightedarithmetic mean and weighted harmonic mean for 119901 = 1 and119901 = minus1 respectivelyDefinition 9 (weighted 120572-geometric mean) Given a set ofpositive reals 1199091 1199092 119909119899 and corresponding weights1199081 1199082 119908119899 then the weighted 120572-geometric mean 120572 isdefined by
120572 = 119899
120572prod119894=1
119909119894(119908119894sum
119899119894=1 119908119894)120572 = 119909(1199081(1199081+sdotsdotsdot+119908119899))1205721
times 119909(1199082(1199081+sdotsdotsdot+119908119899))1205722 times sdot sdot sdot times 119909(119908119899(1199081+sdotsdotsdot+119908119899))120572119899
= 120572[120572minus1 (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [120572minus1 (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [120572minus1 (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(23)
Note that if all the weights are equal the weighted 120572-geometric mean is the same as the 120572-geometric meanTaking 120572 = exp and 120572 = 119902119901 theweighted 120572-geometricmeancan be written for the weights 1199081 1199082 119908119899 as follows
exp = exp [ln (1199091)]1199081(1199081+sdotsdotsdot+119908119899) [ln (1199092)]1199082(1199081+sdotsdotsdot+119908119899) [ln (119909119899)]119908119899(1199081+sdotsdotsdot+119908119899)
(119909119899 gt 1) 119901 = 1199091199011199081(1199081+sdotsdotsdot+119908119899)1 1199091199011199082(1199081+sdotsdotsdot+119908119899)2 119909119901119908119899(1199081+sdotsdotsdot+119908119899)119899 1119901 = ( 119899prod
119894=1
119909119908119894119894 )1sum119899119894=1 119908119894
(24)
exp and 119901 are called weighted multiplicative geometricmean and weighted 119902-geometric mean Also we have 119901 =exp for all 119909119899 gt 1Definition 10 (weighted 120572-harmonic mean) If a set 11990811199082 119908119899 of weights is associated with the data set1199091 1199092 119909119899 then the weighted 120572-harmonic mean isdefined by
120572 = 119899
120572sum119896=1
119896 119899120572sum119896=1
(119896 119909119896)= (1 + sdot sdot sdot + 119899) (1 1199091 + 2 1199092 + sdot sdot sdot + 119899 119909119899)= 120572 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081120572minus1 (1199091) + 1199082120572minus1 (1199092) + sdot sdot sdot + 119908119899120572minus1 (119909119899)
(25)
Taking 120572 = exp and 120572 = 119902119901 the weighted 120572-harmonicmean with the weights 1199081 1199082 119908119899 can be written asfollows
exp
= exp 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 ln1199091 + 1199082 ln1199092 + sdot sdot sdot + 119908119899 ln119909119899 (119909119899 gt 1)
119901= 1199081 + 1199082 + sdot sdot sdot + 1199081198991199081 (1199091)119901 + 1199082 (1199092)119901 + sdot sdot sdot + 119908119899 (119909119899)119901
1119901
(26)
exp and 119901 are called multiplicative weighted harmonicmean and weighted 119901-geometric mean respectively It isobvious that 119901 is reduced to ordinary weighted harmonicmean and ordinary weighted arithmetic mean for 119901 = 1 and119901 = minus1 respectively
6 International Journal of Analysis
Table 1 Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means
Weight Data 119901 119860119901 119866119901 119867119901 119901 119901 119901 119866exp 119867exp exp exp1199081 = 2 1199091 = 15 10 2750 2474 2264 3282 2988 2727 2402 2336 2905 28261199082 = 5 1199092 = 20 01 2499 2474 2450 3016 2988 2959 2402 2336 2905 28261199083 = 7 1199093 = 25 20 3061 2474 2114 3577 2988 2521 2402 2336 2905 28261199084 = 9 1199094 = 50 50 3822 2474 1873 4169 2988 2155 2402 2336 2905 2826
In Table 1 the non-Newtonian means are obtained byusing different generating functions For 120572 = 119902119901 the119901-means119860119901 119866119901 and119867119901 are reduced to ordinary arithmeticmean geo-metric mean and harmonic mean respectively In particularsome changes are observed for each value of 119860119901 119866119901 and119867119901means depending on the choice of 119901 As shown in the tablefor increasing values of 119901 the 119901-arithmetic mean 119860119901 and itsweighted form 119901 increase in particular 119901 tends to infin andthese means converge to the value of max119909119899 Converselyfor increasing values of 119901 the 119901-harmonic mean 119867119901 andits weighted forms 119901 decrease In particular these meansconverge to the value of min119909119899 as 119901 rarr infin Dependingon the choice of 119901 weighted forms 119901 and 119901 can beincreased or decreased without changing any weights Forthis reason this approach brings a new perspective to theconcept of classical (weighted) mean Moreover when wecompare 119867exp and ordinary harmonic mean in Table 1 wealso see that ordinary harmonic mean is smaller than 119867expOn the contrary119860exp and119866exp are smaller than their classicalforms 119860119901 and 119866119901 for 119901 = 1 Therefore we assert thatthe values of 119866exp 119867exp exp and exp should be evaluatedsatisfactorily
Corollary 11 Consider 119899 positive real numbers 1199091 1199092 119909119899Then the conditions119867120572 lt 119866120572 lt 119860120572 and 120572 lt 120572 lt 120572 holdwhen 120572 = exp for all 119909119899 gt 1 and 120572 = 119902119901 for all 119901 isin R+
4 Non-Newtonian Convexity
In this section the notion of non-Newtonian convex (lowast-convex) functions will be given by using different genera-tors Furthermore the relationships between lowast-convexity andnon-Newtonian weighted mean will be determined
Definition 12 (generalized lowast-convex function) Let 119868120572 be aninterval in R120572 Then 119891 119868120572 rarr R120573 is said to be lowast-convex if
119891 (1205821 times 119909 + 1205822 times 119910) le 1205831 times 119891 (119909) + 1205832 times 119891 (119910) (27)
holds where 1205821 + 1205822 = 1 and 1205831 + 1205832 = 1 for all 1205821 1205822 isin[0 1] and 1205831 1205832 isin [0 1] Therefore by combining this withthe generators 120572 and 120573 we deduce that
119891 (120572 120572minus1 (1205821) 120572minus1 (119909) + 120572minus1 (1205822) 120572minus1 (119910))le 120573 120573minus1 (1205831) 120573minus1119891 (119909) + 120573minus1 (1205832) 120573minus1119891 (119910) (28)
If (28) is strict for all 119909 = 119910 then 119891 is said to be strictly lowast-convex If the inequality in (28) is reversed then 119891 is said to
be lowast-concave On the other hand the inclusion (28) can bewritten with respect to the weighted 120572-arithmetic mean in(21) as follows
119891 (120572 119909 119910) le 120573 119891 (119909) 119891 (119910) (29)
Remark 13 We remark that the definition of lowast-convexity in(27) can be evaluated by non-Newtonian coordinate systeminvolving lowast-lines (see Definition 3) For 120572 = 120573 = 119868 the lowast-lines are straight lines in two-dimensional Euclidean spaceFor this reason we say that almost all the properties ofordinary Cartesian coordinate system will be valid for non-Newtonian coordinate system under lowast-arithmetic
Also depending on the choice of generator functionsthe definition of lowast-convexity in (27) can be interpreted asfollows
Case 1 (a) If we take 120572 = 120573 = exp and 1205821 = 1205831 1205822 = 1205832 in(28) then
119891 (119909ln1205821119910ln1205822) le 119891 (119909)ln1205821 119891 (119910)ln1205822 (1205821 1205822 isin [1 119890])
(30)
where 12058211205822 = 119890 holds and 119891 119868exp rarr Rexp = (0infin) iscalled bigeometric (usually known as multiplicative) convexfunction (cf [2]) Equivalently 119891 is bigeometric convex ifand only if log119891(119909) is an ordinary convex function
(b) For 120572 = exp and 120573 = 119868 we have119891 (119909ln1205821119910ln1205822) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 isin [1 119890] 1205831 1205832 isin [0 1]) (31)
where 12058211205822 = 119890 and 1205831 +1205832 = 1 In this case the function 119891 119868exp rarr R is called geometric convex function Everygeometric convex (usually known as log-convex) function isalso convex (cf [2])
(c) Taking 120572 = 119868 and 120573 = exp one obtains
119891 (1205821119909 + 1205822119910) le 119891 (119909)ln 1205831 119891 (119910)ln 1205832 (1205831 1205832 isin [1 119890] 1205821 1205822 isin [0 1]) (32)
where 12058311205832 = 119890 and 1205821 + 1205822 = 1 and 119891 119868 rarr Rexp is calledanageometric convex function
Case 2 (a) If 120572 = 120573 = 119902119901 in (28) then
119891(((1205821119909)119901 + (1205822119910)119901)1119901)le ((1205821119891 (119909))119901 + (1205822119891 (119910))119901)1119901 (119901 isin R
+) (33)
International Journal of Analysis 7
where 1205821 1205822 isin [0 1] 1205821199011 + 1205821199012 = 1 and 119891 119868119902119901 rarr R119902119901 iscalled 119876119876-convex function
(b) For 120572 = 119902119901 and 120573 = 119868 we write that119891(((1205821119909)119901 + (1205822119910)119901)1119901) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 1205831 1205832 isin [0 1]) (34)
where 1205821199011 + 1205821199012 = 1 1205831 + 1205832 = 1 and 119891 119868119902119901 rarr R is called119876119868-convex function(c) For 120572 = 119868 and 120573 = 119902119901 we obtain that
119891 (1205821119909 + 1205822119910) le ((1205831119891 (119909))119901 + (1205832119891 (119910))119901)1119901 (1205821 1205822 1205831 1205832 isin [0 1])
(35)
where 1205831199011 +1205831199012 = 1 1205821 +1205822 = 1 and 119891 119868 rarr R119902119901 is called 119868119876-convex function
The lowast-convexity of a function 119891 119868120572 rarr R120573 means geo-metrically that the lowast-points of the graph of 119891 are under thechord joining the endpoints (119886 119891(119886)) and (119887 119891(119887)) on non-Newtonian coordinate system for every 119886 119887 isin 119868120572 By takinginto account the definition of lowast-slope in Definition 3 we have
(119891 (119909) minus 119891 (119886)) 120580 (119909 minus 119886)le (119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886) (36)
which implies
119891 (119909) le 119891 (119886) + ((119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886))times 120580 (119909 minus 119886) (37)
for all 119909 isin [ ]On the other hand (37) means that if 119875119876 and 119877 are any
three lowast-points on the graph of 119891 with 119876 between 119875 and 119877then 119876 is on or below chord 119875119877 In terms of lowast-slope it isequivalent to
119898lowast (119875119876) le119898lowast (119875119877) le119898lowast (119876119877) (38)
with strict inequalities when 119891 is strictly lowast-convexNow to avoid the repetition of the similar statements we
give some necessary theorems and lemmas
Lemma 14 (Jensenrsquos inequality) A 120573-real-valued function 119891defined on an interval 119868120572 is lowast-convex if and only if
119891( 119899
120572sum119896=1
120582119896 times 119909119896) le 119899
120573sum119896=1
120583119896 times 119891 (119909119896) (39)
holds where 120572sum119899119896=1 120582119896 = 1 and 120573sum119899119896=1 120583119896 = 1 for all 120582119899 isin [0 1]and 120583119899 isin [0 1]Proof The proof is straightforward hence omitted
Theorem 15 Let 119891 119868120572 rarr R120573 be a lowast-continuous functionThen 119891 is lowast-convex if and only if 119891 is midpoint lowast-convexthat is
1199091 1199092 isin 119868120572 implies
119891 (119860120572 1199091 1199092) le 119860120573 119891 (1199091) 119891 (1199092) (40)
Proof The proof can be easily obtained using the inequality(39) in Lemma 14
Theorem16 (cf [2]) Let 119891 119868exp rarr Rexp be alowast-differentiablefunction (see [19]) on a subinterval 119868exp sube (0infin) Then thefollowing assertions are equivalent
(i) 119891 is bigeometric convex (concave)(ii) The function 119891lowast(119909) is increasing (decreasing)
Corollary 17 A positive 120573-real-valued function 119891 defined onan interval 119868exp is bigeometric convex if and only if
119891 (119909ln12058211 119909ln12058222 119909ln120582119899119899 )le 119891 (1199091)ln1205821 119891 (1199092)ln1205822 119891 (119909119899)ln120582119899
(41)
holds where prod119899119896=1120582119896 = 119890 for all 1199091 1199092 119909119899 isin 119868exp and1205821 1205822 120582119899 isin [1 119890] Besides we have119891 (exp 1199091 1199092 119909119899)le exp 119891 (1199091) 119891 (1199092) 119891 (119909119899) (42)
Corollary 18 A 120573-real-valued function119891 defined on an inter-val 119868119902119901 is 119876119876-convex if and only if
119891[((12058211199091)119901 + (12058221199092)119901 + sdot sdot sdot + (120582119899119909119899)119901)1119901]le ((1205821119891 (1199091))119901 + (1205822119891 (1199092))119901 + sdot sdot sdot+ (120582119899119891 (119909119899))119901)1119901
(43)
holds where sum119899119896=1 120582119901119896 = 1 for all 1199091 1199092 119909119899 isin 119868119902119901 and1205821 1205822 120582119899 isin [0 1] Thus we have
119891 (119901 1199091 1199092 119909119899)le 119901 119891 (1199091) 119891 (1199092) 119891 (119909119899) (119901 isin R
+) (44)
5 An Application of Multiplicative Continuity
In this section based on the definition of bigeometric convexfunction and multiplicative continuity we get an analogue ofordinary Lipschitz condition on any closed interval
Let 119891 be a bigeometric (multiplicative) convex functionand finite on a closed interval [119909 119910] sub R+ It is obvious that119891is bounded from above by 119872 = max119891(119909) 119891(119910) sincefor any 119911 = 1199091205821199101minus120582 in the interval 119891(119911) le 119891(119909)120582119891(119910)1minus120582 for
8 International Journal of Analysis
120582 isin [1 119890] It is also bounded from below as we see by writingan arbitrary point in the form 119905radic119909119910 for 119905 isin R+ Then
1198912 (radic119909119910) le 119891 (119905radic119909119910)119891(radic119909119910119905 ) (45)
Using 119872 as the upper bound 119891(radic119909119910119905) we obtain119891 (119905radic119909119910) ge 11198721198912 (radic119909119910) = 119898 (46)
Thus a bigeometric convex functionmay not be continuous atthe boundary points of its domainWewill prove that for anyclosed subinterval [119909 119910] of the interior of the domain thereis a constant 119870 gt 0 so that for any two points 119886 119887 isin [119909 119910] subR+
119891 (119886)119891 (119887) le (119886119887)119870 (47)
A function that satisfies (47) for some 119870 and all 119886 and 119887 inan interval is said to satisfy bigeometric Lipschitz conditionon the interval
Theorem 19 Suppose that 119891 119868 rarr R+ is multiplicative con-vex Then 119891 satisfies the multiplicative Lipschitz condition onany closed interval [119909 119910] sub R+ contained in the interior 1198680 of119868 that is 119891 is continuous on 1198680Proof Take 120576 gt 1 so that [119909120576 119910120576] isin 119868 and let 119898 and 119872 bethe lower and upper bounds for 119891 on [119909120576 119910120576] If 119903 and 119904 aredistinct points of [119909 119910] with 119904 gt 119903 set
119911 = 119904120576120582 = ( 119904119903)
1 ln(120576119904119903) (120582 isin (1 119890)) (48)
Then 119911 isin [119909120576 119910120576] and 119904 = 119911ln1205821199031minusln120582 and we obtain
119891 (119904) le 119891 (119911)ln120582 119891 (119903)1minusln120582 = (119891 (119911)119891 (119903) )ln120582 119891 (119903) (49)
which yields
ln(119891 (119904)119891 (119903)) le ln 120582 ln(119891 (119911)119891 (119903) )lt ln( 119904119903)
1 ln(120576)ln(119872119898 )
119891 (119904)119891 (119903) le ( 119904119903)ln(119872119898) ln(120576)
(50)
where 119870 = ln(119872119898) ln(120576) gt 0 Since the points 119903 119904 isin [119909 119910]are arbitrary we get 119891 that satisfies a multiplicative Lipschitzcondition The remaining part can be obtained in the similarway by taking 119904 lt 119903 and 119911 = 119904120576 Finally 119891 is continuoussince [119909 119910] is arbitrary in 1198680
6 Concluding Remarks
Although all arithmetics are isomorphic only by distinguish-ing among them do we obtain suitable tools for construct-ing all the non-Newtonian calculi But the usefulness ofarithmetic is not limited to the construction of calculi webelieve there is a more fundamental reason for consideringalternative arithmetics theymay also be helpful in developingand understanding new systems of measurement that couldyield simpler physical laws
In this paper it was shown that due to the choice ofgenerator function 119860119901 119866119901 and 119867119901 means are reduced toordinary arithmetic geometric and harmonic mean respec-tively As shown in Table 1 for increasing values of 119901 119860119901and 119901 means increase especially 119901 rarr infin these meansconverge to the value of max119909119899 Conversely for increasingvalues of 119901 119867119901 and 119901 means decrease especially 119901 rarr infinthese means converge to the value of min119909119899 Additionallywe give some new definitions regarding convex functionswhich are plotted on the non-Newtonian coordinate systemObviously for different generator functions one can obtainsome new geometrical interpretations of convex functionsOur future works will include the most famous HermiteHadamard inequality for the class of lowast-convex functionsCompeting Interests
The authors declare that they have no competing interests
References
[1] C Niculescu and L-E Persson Convex Functions and TheirApplications A Contemporary Approach Springer Berlin Ger-many 2006
[2] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000
[3] R Webster Convexity Oxford University Press New York NYUSA 1995
[4] J Banas and A Ben Amar ldquoMeasures of noncompactness inlocally convex spaces and fixed point theory for the sum oftwo operators on unbounded convex setsrdquo CommentationesMathematicae Universitatis Carolinae vol 54 no 1 pp 21ndash402013
[5] J Matkowski ldquoGeneralized weighted and quasi-arithmeticmeansrdquo Aequationes Mathematicae vol 79 no 3 pp 203ndash2122010
[6] D Głazowska and J Matkowski ldquoAn invariance of geometricmean with respect to Lagrangian meansrdquo Journal of Mathemat-ical Analysis and Applications vol 331 no 2 pp 1187ndash1199 2007
[7] J Matkowski ldquoGeneralized weighted quasi-arithmetic meansand the Kolmogorov-Nagumo theoremrdquo Colloquium Mathe-maticum vol 133 no 1 pp 35ndash49 2013
[8] N Merentes and K Nikodem ldquoRemarks on strongly convexfunctionsrdquo Aequationes Mathematicae vol 80 no 1-2 pp 193ndash199 2010
[9] M Grossman Bigeometric Calculus Archimedes FoundationBox 240 Rockport Mass USA 1983
[10] M Grossman and R Katz Non-Newtonian Calculus NewtonInstitute 1978
International Journal of Analysis 9
[11] M Grossman The First Nonlinear System of Differential andIntegral Calculus 1979
[12] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008
[13] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007
[14] E Mısırlı and Y Gurefe ldquoMultiplicative adams-bashforth-moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011
[15] A F Cakmak and F Basar ldquoSome new results on sequencespaces with respect to non-Newtonian calculusrdquo Journal ofInequalities and Applications vol 2012 article 228 2012
[16] A F Cakmak and F Basar ldquoCertain spaces of functions overthe field of non-Newtonian complex numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 236124 12 pages 2014
[17] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013
[18] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014
[19] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015
[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014
[21] U Kadak ldquoNon-Newtonian fuzzy numbers and related applica-tionsrdquo Iranian Journal of Fuzzy Systems vol 12 no 5 pp 117ndash1372015
[22] Y Gurefe U Kadak E Misirli and A Kurdi ldquoA new look atthe classical sequence spaces by using multiplicative calculusrdquoUniversity Politehnica of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 78 no 2 pp 9ndash20 2016
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Stochastic AnalysisInternational Journal of
6 International Journal of Analysis
Table 1 Comparison of the non-Newtonian (weighted) means and ordinary (weighted) means
Weight Data 119901 119860119901 119866119901 119867119901 119901 119901 119901 119866exp 119867exp exp exp1199081 = 2 1199091 = 15 10 2750 2474 2264 3282 2988 2727 2402 2336 2905 28261199082 = 5 1199092 = 20 01 2499 2474 2450 3016 2988 2959 2402 2336 2905 28261199083 = 7 1199093 = 25 20 3061 2474 2114 3577 2988 2521 2402 2336 2905 28261199084 = 9 1199094 = 50 50 3822 2474 1873 4169 2988 2155 2402 2336 2905 2826
In Table 1 the non-Newtonian means are obtained byusing different generating functions For 120572 = 119902119901 the119901-means119860119901 119866119901 and119867119901 are reduced to ordinary arithmeticmean geo-metric mean and harmonic mean respectively In particularsome changes are observed for each value of 119860119901 119866119901 and119867119901means depending on the choice of 119901 As shown in the tablefor increasing values of 119901 the 119901-arithmetic mean 119860119901 and itsweighted form 119901 increase in particular 119901 tends to infin andthese means converge to the value of max119909119899 Converselyfor increasing values of 119901 the 119901-harmonic mean 119867119901 andits weighted forms 119901 decrease In particular these meansconverge to the value of min119909119899 as 119901 rarr infin Dependingon the choice of 119901 weighted forms 119901 and 119901 can beincreased or decreased without changing any weights Forthis reason this approach brings a new perspective to theconcept of classical (weighted) mean Moreover when wecompare 119867exp and ordinary harmonic mean in Table 1 wealso see that ordinary harmonic mean is smaller than 119867expOn the contrary119860exp and119866exp are smaller than their classicalforms 119860119901 and 119866119901 for 119901 = 1 Therefore we assert thatthe values of 119866exp 119867exp exp and exp should be evaluatedsatisfactorily
Corollary 11 Consider 119899 positive real numbers 1199091 1199092 119909119899Then the conditions119867120572 lt 119866120572 lt 119860120572 and 120572 lt 120572 lt 120572 holdwhen 120572 = exp for all 119909119899 gt 1 and 120572 = 119902119901 for all 119901 isin R+
4 Non-Newtonian Convexity
In this section the notion of non-Newtonian convex (lowast-convex) functions will be given by using different genera-tors Furthermore the relationships between lowast-convexity andnon-Newtonian weighted mean will be determined
Definition 12 (generalized lowast-convex function) Let 119868120572 be aninterval in R120572 Then 119891 119868120572 rarr R120573 is said to be lowast-convex if
119891 (1205821 times 119909 + 1205822 times 119910) le 1205831 times 119891 (119909) + 1205832 times 119891 (119910) (27)
holds where 1205821 + 1205822 = 1 and 1205831 + 1205832 = 1 for all 1205821 1205822 isin[0 1] and 1205831 1205832 isin [0 1] Therefore by combining this withthe generators 120572 and 120573 we deduce that
119891 (120572 120572minus1 (1205821) 120572minus1 (119909) + 120572minus1 (1205822) 120572minus1 (119910))le 120573 120573minus1 (1205831) 120573minus1119891 (119909) + 120573minus1 (1205832) 120573minus1119891 (119910) (28)
If (28) is strict for all 119909 = 119910 then 119891 is said to be strictly lowast-convex If the inequality in (28) is reversed then 119891 is said to
be lowast-concave On the other hand the inclusion (28) can bewritten with respect to the weighted 120572-arithmetic mean in(21) as follows
119891 (120572 119909 119910) le 120573 119891 (119909) 119891 (119910) (29)
Remark 13 We remark that the definition of lowast-convexity in(27) can be evaluated by non-Newtonian coordinate systeminvolving lowast-lines (see Definition 3) For 120572 = 120573 = 119868 the lowast-lines are straight lines in two-dimensional Euclidean spaceFor this reason we say that almost all the properties ofordinary Cartesian coordinate system will be valid for non-Newtonian coordinate system under lowast-arithmetic
Also depending on the choice of generator functionsthe definition of lowast-convexity in (27) can be interpreted asfollows
Case 1 (a) If we take 120572 = 120573 = exp and 1205821 = 1205831 1205822 = 1205832 in(28) then
119891 (119909ln1205821119910ln1205822) le 119891 (119909)ln1205821 119891 (119910)ln1205822 (1205821 1205822 isin [1 119890])
(30)
where 12058211205822 = 119890 holds and 119891 119868exp rarr Rexp = (0infin) iscalled bigeometric (usually known as multiplicative) convexfunction (cf [2]) Equivalently 119891 is bigeometric convex ifand only if log119891(119909) is an ordinary convex function
(b) For 120572 = exp and 120573 = 119868 we have119891 (119909ln1205821119910ln1205822) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 isin [1 119890] 1205831 1205832 isin [0 1]) (31)
where 12058211205822 = 119890 and 1205831 +1205832 = 1 In this case the function 119891 119868exp rarr R is called geometric convex function Everygeometric convex (usually known as log-convex) function isalso convex (cf [2])
(c) Taking 120572 = 119868 and 120573 = exp one obtains
119891 (1205821119909 + 1205822119910) le 119891 (119909)ln 1205831 119891 (119910)ln 1205832 (1205831 1205832 isin [1 119890] 1205821 1205822 isin [0 1]) (32)
where 12058311205832 = 119890 and 1205821 + 1205822 = 1 and 119891 119868 rarr Rexp is calledanageometric convex function
Case 2 (a) If 120572 = 120573 = 119902119901 in (28) then
119891(((1205821119909)119901 + (1205822119910)119901)1119901)le ((1205821119891 (119909))119901 + (1205822119891 (119910))119901)1119901 (119901 isin R
+) (33)
International Journal of Analysis 7
where 1205821 1205822 isin [0 1] 1205821199011 + 1205821199012 = 1 and 119891 119868119902119901 rarr R119902119901 iscalled 119876119876-convex function
(b) For 120572 = 119902119901 and 120573 = 119868 we write that119891(((1205821119909)119901 + (1205822119910)119901)1119901) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 1205831 1205832 isin [0 1]) (34)
where 1205821199011 + 1205821199012 = 1 1205831 + 1205832 = 1 and 119891 119868119902119901 rarr R is called119876119868-convex function(c) For 120572 = 119868 and 120573 = 119902119901 we obtain that
119891 (1205821119909 + 1205822119910) le ((1205831119891 (119909))119901 + (1205832119891 (119910))119901)1119901 (1205821 1205822 1205831 1205832 isin [0 1])
(35)
where 1205831199011 +1205831199012 = 1 1205821 +1205822 = 1 and 119891 119868 rarr R119902119901 is called 119868119876-convex function
The lowast-convexity of a function 119891 119868120572 rarr R120573 means geo-metrically that the lowast-points of the graph of 119891 are under thechord joining the endpoints (119886 119891(119886)) and (119887 119891(119887)) on non-Newtonian coordinate system for every 119886 119887 isin 119868120572 By takinginto account the definition of lowast-slope in Definition 3 we have
(119891 (119909) minus 119891 (119886)) 120580 (119909 minus 119886)le (119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886) (36)
which implies
119891 (119909) le 119891 (119886) + ((119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886))times 120580 (119909 minus 119886) (37)
for all 119909 isin [ ]On the other hand (37) means that if 119875119876 and 119877 are any
three lowast-points on the graph of 119891 with 119876 between 119875 and 119877then 119876 is on or below chord 119875119877 In terms of lowast-slope it isequivalent to
119898lowast (119875119876) le119898lowast (119875119877) le119898lowast (119876119877) (38)
with strict inequalities when 119891 is strictly lowast-convexNow to avoid the repetition of the similar statements we
give some necessary theorems and lemmas
Lemma 14 (Jensenrsquos inequality) A 120573-real-valued function 119891defined on an interval 119868120572 is lowast-convex if and only if
119891( 119899
120572sum119896=1
120582119896 times 119909119896) le 119899
120573sum119896=1
120583119896 times 119891 (119909119896) (39)
holds where 120572sum119899119896=1 120582119896 = 1 and 120573sum119899119896=1 120583119896 = 1 for all 120582119899 isin [0 1]and 120583119899 isin [0 1]Proof The proof is straightforward hence omitted
Theorem 15 Let 119891 119868120572 rarr R120573 be a lowast-continuous functionThen 119891 is lowast-convex if and only if 119891 is midpoint lowast-convexthat is
1199091 1199092 isin 119868120572 implies
119891 (119860120572 1199091 1199092) le 119860120573 119891 (1199091) 119891 (1199092) (40)
Proof The proof can be easily obtained using the inequality(39) in Lemma 14
Theorem16 (cf [2]) Let 119891 119868exp rarr Rexp be alowast-differentiablefunction (see [19]) on a subinterval 119868exp sube (0infin) Then thefollowing assertions are equivalent
(i) 119891 is bigeometric convex (concave)(ii) The function 119891lowast(119909) is increasing (decreasing)
Corollary 17 A positive 120573-real-valued function 119891 defined onan interval 119868exp is bigeometric convex if and only if
119891 (119909ln12058211 119909ln12058222 119909ln120582119899119899 )le 119891 (1199091)ln1205821 119891 (1199092)ln1205822 119891 (119909119899)ln120582119899
(41)
holds where prod119899119896=1120582119896 = 119890 for all 1199091 1199092 119909119899 isin 119868exp and1205821 1205822 120582119899 isin [1 119890] Besides we have119891 (exp 1199091 1199092 119909119899)le exp 119891 (1199091) 119891 (1199092) 119891 (119909119899) (42)
Corollary 18 A 120573-real-valued function119891 defined on an inter-val 119868119902119901 is 119876119876-convex if and only if
119891[((12058211199091)119901 + (12058221199092)119901 + sdot sdot sdot + (120582119899119909119899)119901)1119901]le ((1205821119891 (1199091))119901 + (1205822119891 (1199092))119901 + sdot sdot sdot+ (120582119899119891 (119909119899))119901)1119901
(43)
holds where sum119899119896=1 120582119901119896 = 1 for all 1199091 1199092 119909119899 isin 119868119902119901 and1205821 1205822 120582119899 isin [0 1] Thus we have
119891 (119901 1199091 1199092 119909119899)le 119901 119891 (1199091) 119891 (1199092) 119891 (119909119899) (119901 isin R
+) (44)
5 An Application of Multiplicative Continuity
In this section based on the definition of bigeometric convexfunction and multiplicative continuity we get an analogue ofordinary Lipschitz condition on any closed interval
Let 119891 be a bigeometric (multiplicative) convex functionand finite on a closed interval [119909 119910] sub R+ It is obvious that119891is bounded from above by 119872 = max119891(119909) 119891(119910) sincefor any 119911 = 1199091205821199101minus120582 in the interval 119891(119911) le 119891(119909)120582119891(119910)1minus120582 for
8 International Journal of Analysis
120582 isin [1 119890] It is also bounded from below as we see by writingan arbitrary point in the form 119905radic119909119910 for 119905 isin R+ Then
1198912 (radic119909119910) le 119891 (119905radic119909119910)119891(radic119909119910119905 ) (45)
Using 119872 as the upper bound 119891(radic119909119910119905) we obtain119891 (119905radic119909119910) ge 11198721198912 (radic119909119910) = 119898 (46)
Thus a bigeometric convex functionmay not be continuous atthe boundary points of its domainWewill prove that for anyclosed subinterval [119909 119910] of the interior of the domain thereis a constant 119870 gt 0 so that for any two points 119886 119887 isin [119909 119910] subR+
119891 (119886)119891 (119887) le (119886119887)119870 (47)
A function that satisfies (47) for some 119870 and all 119886 and 119887 inan interval is said to satisfy bigeometric Lipschitz conditionon the interval
Theorem 19 Suppose that 119891 119868 rarr R+ is multiplicative con-vex Then 119891 satisfies the multiplicative Lipschitz condition onany closed interval [119909 119910] sub R+ contained in the interior 1198680 of119868 that is 119891 is continuous on 1198680Proof Take 120576 gt 1 so that [119909120576 119910120576] isin 119868 and let 119898 and 119872 bethe lower and upper bounds for 119891 on [119909120576 119910120576] If 119903 and 119904 aredistinct points of [119909 119910] with 119904 gt 119903 set
119911 = 119904120576120582 = ( 119904119903)
1 ln(120576119904119903) (120582 isin (1 119890)) (48)
Then 119911 isin [119909120576 119910120576] and 119904 = 119911ln1205821199031minusln120582 and we obtain
119891 (119904) le 119891 (119911)ln120582 119891 (119903)1minusln120582 = (119891 (119911)119891 (119903) )ln120582 119891 (119903) (49)
which yields
ln(119891 (119904)119891 (119903)) le ln 120582 ln(119891 (119911)119891 (119903) )lt ln( 119904119903)
1 ln(120576)ln(119872119898 )
119891 (119904)119891 (119903) le ( 119904119903)ln(119872119898) ln(120576)
(50)
where 119870 = ln(119872119898) ln(120576) gt 0 Since the points 119903 119904 isin [119909 119910]are arbitrary we get 119891 that satisfies a multiplicative Lipschitzcondition The remaining part can be obtained in the similarway by taking 119904 lt 119903 and 119911 = 119904120576 Finally 119891 is continuoussince [119909 119910] is arbitrary in 1198680
6 Concluding Remarks
Although all arithmetics are isomorphic only by distinguish-ing among them do we obtain suitable tools for construct-ing all the non-Newtonian calculi But the usefulness ofarithmetic is not limited to the construction of calculi webelieve there is a more fundamental reason for consideringalternative arithmetics theymay also be helpful in developingand understanding new systems of measurement that couldyield simpler physical laws
In this paper it was shown that due to the choice ofgenerator function 119860119901 119866119901 and 119867119901 means are reduced toordinary arithmetic geometric and harmonic mean respec-tively As shown in Table 1 for increasing values of 119901 119860119901and 119901 means increase especially 119901 rarr infin these meansconverge to the value of max119909119899 Conversely for increasingvalues of 119901 119867119901 and 119901 means decrease especially 119901 rarr infinthese means converge to the value of min119909119899 Additionallywe give some new definitions regarding convex functionswhich are plotted on the non-Newtonian coordinate systemObviously for different generator functions one can obtainsome new geometrical interpretations of convex functionsOur future works will include the most famous HermiteHadamard inequality for the class of lowast-convex functionsCompeting Interests
The authors declare that they have no competing interests
References
[1] C Niculescu and L-E Persson Convex Functions and TheirApplications A Contemporary Approach Springer Berlin Ger-many 2006
[2] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000
[3] R Webster Convexity Oxford University Press New York NYUSA 1995
[4] J Banas and A Ben Amar ldquoMeasures of noncompactness inlocally convex spaces and fixed point theory for the sum oftwo operators on unbounded convex setsrdquo CommentationesMathematicae Universitatis Carolinae vol 54 no 1 pp 21ndash402013
[5] J Matkowski ldquoGeneralized weighted and quasi-arithmeticmeansrdquo Aequationes Mathematicae vol 79 no 3 pp 203ndash2122010
[6] D Głazowska and J Matkowski ldquoAn invariance of geometricmean with respect to Lagrangian meansrdquo Journal of Mathemat-ical Analysis and Applications vol 331 no 2 pp 1187ndash1199 2007
[7] J Matkowski ldquoGeneralized weighted quasi-arithmetic meansand the Kolmogorov-Nagumo theoremrdquo Colloquium Mathe-maticum vol 133 no 1 pp 35ndash49 2013
[8] N Merentes and K Nikodem ldquoRemarks on strongly convexfunctionsrdquo Aequationes Mathematicae vol 80 no 1-2 pp 193ndash199 2010
[9] M Grossman Bigeometric Calculus Archimedes FoundationBox 240 Rockport Mass USA 1983
[10] M Grossman and R Katz Non-Newtonian Calculus NewtonInstitute 1978
International Journal of Analysis 9
[11] M Grossman The First Nonlinear System of Differential andIntegral Calculus 1979
[12] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008
[13] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007
[14] E Mısırlı and Y Gurefe ldquoMultiplicative adams-bashforth-moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011
[15] A F Cakmak and F Basar ldquoSome new results on sequencespaces with respect to non-Newtonian calculusrdquo Journal ofInequalities and Applications vol 2012 article 228 2012
[16] A F Cakmak and F Basar ldquoCertain spaces of functions overthe field of non-Newtonian complex numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 236124 12 pages 2014
[17] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013
[18] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014
[19] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015
[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014
[21] U Kadak ldquoNon-Newtonian fuzzy numbers and related applica-tionsrdquo Iranian Journal of Fuzzy Systems vol 12 no 5 pp 117ndash1372015
[22] Y Gurefe U Kadak E Misirli and A Kurdi ldquoA new look atthe classical sequence spaces by using multiplicative calculusrdquoUniversity Politehnica of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 78 no 2 pp 9ndash20 2016
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 7
where 1205821 1205822 isin [0 1] 1205821199011 + 1205821199012 = 1 and 119891 119868119902119901 rarr R119902119901 iscalled 119876119876-convex function
(b) For 120572 = 119902119901 and 120573 = 119868 we write that119891(((1205821119909)119901 + (1205822119910)119901)1119901) le 1205831119891 (119909) + 1205832119891 (119910)
(1205821 1205822 1205831 1205832 isin [0 1]) (34)
where 1205821199011 + 1205821199012 = 1 1205831 + 1205832 = 1 and 119891 119868119902119901 rarr R is called119876119868-convex function(c) For 120572 = 119868 and 120573 = 119902119901 we obtain that
119891 (1205821119909 + 1205822119910) le ((1205831119891 (119909))119901 + (1205832119891 (119910))119901)1119901 (1205821 1205822 1205831 1205832 isin [0 1])
(35)
where 1205831199011 +1205831199012 = 1 1205821 +1205822 = 1 and 119891 119868 rarr R119902119901 is called 119868119876-convex function
The lowast-convexity of a function 119891 119868120572 rarr R120573 means geo-metrically that the lowast-points of the graph of 119891 are under thechord joining the endpoints (119886 119891(119886)) and (119887 119891(119887)) on non-Newtonian coordinate system for every 119886 119887 isin 119868120572 By takinginto account the definition of lowast-slope in Definition 3 we have
(119891 (119909) minus 119891 (119886)) 120580 (119909 minus 119886)le (119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886) (36)
which implies
119891 (119909) le 119891 (119886) + ((119891 (119887) minus 119891 (119886)) 120580 (119887 minus 119886))times 120580 (119909 minus 119886) (37)
for all 119909 isin [ ]On the other hand (37) means that if 119875119876 and 119877 are any
three lowast-points on the graph of 119891 with 119876 between 119875 and 119877then 119876 is on or below chord 119875119877 In terms of lowast-slope it isequivalent to
119898lowast (119875119876) le119898lowast (119875119877) le119898lowast (119876119877) (38)
with strict inequalities when 119891 is strictly lowast-convexNow to avoid the repetition of the similar statements we
give some necessary theorems and lemmas
Lemma 14 (Jensenrsquos inequality) A 120573-real-valued function 119891defined on an interval 119868120572 is lowast-convex if and only if
119891( 119899
120572sum119896=1
120582119896 times 119909119896) le 119899
120573sum119896=1
120583119896 times 119891 (119909119896) (39)
holds where 120572sum119899119896=1 120582119896 = 1 and 120573sum119899119896=1 120583119896 = 1 for all 120582119899 isin [0 1]and 120583119899 isin [0 1]Proof The proof is straightforward hence omitted
Theorem 15 Let 119891 119868120572 rarr R120573 be a lowast-continuous functionThen 119891 is lowast-convex if and only if 119891 is midpoint lowast-convexthat is
1199091 1199092 isin 119868120572 implies
119891 (119860120572 1199091 1199092) le 119860120573 119891 (1199091) 119891 (1199092) (40)
Proof The proof can be easily obtained using the inequality(39) in Lemma 14
Theorem16 (cf [2]) Let 119891 119868exp rarr Rexp be alowast-differentiablefunction (see [19]) on a subinterval 119868exp sube (0infin) Then thefollowing assertions are equivalent
(i) 119891 is bigeometric convex (concave)(ii) The function 119891lowast(119909) is increasing (decreasing)
Corollary 17 A positive 120573-real-valued function 119891 defined onan interval 119868exp is bigeometric convex if and only if
119891 (119909ln12058211 119909ln12058222 119909ln120582119899119899 )le 119891 (1199091)ln1205821 119891 (1199092)ln1205822 119891 (119909119899)ln120582119899
(41)
holds where prod119899119896=1120582119896 = 119890 for all 1199091 1199092 119909119899 isin 119868exp and1205821 1205822 120582119899 isin [1 119890] Besides we have119891 (exp 1199091 1199092 119909119899)le exp 119891 (1199091) 119891 (1199092) 119891 (119909119899) (42)
Corollary 18 A 120573-real-valued function119891 defined on an inter-val 119868119902119901 is 119876119876-convex if and only if
119891[((12058211199091)119901 + (12058221199092)119901 + sdot sdot sdot + (120582119899119909119899)119901)1119901]le ((1205821119891 (1199091))119901 + (1205822119891 (1199092))119901 + sdot sdot sdot+ (120582119899119891 (119909119899))119901)1119901
(43)
holds where sum119899119896=1 120582119901119896 = 1 for all 1199091 1199092 119909119899 isin 119868119902119901 and1205821 1205822 120582119899 isin [0 1] Thus we have
119891 (119901 1199091 1199092 119909119899)le 119901 119891 (1199091) 119891 (1199092) 119891 (119909119899) (119901 isin R
+) (44)
5 An Application of Multiplicative Continuity
In this section based on the definition of bigeometric convexfunction and multiplicative continuity we get an analogue ofordinary Lipschitz condition on any closed interval
Let 119891 be a bigeometric (multiplicative) convex functionand finite on a closed interval [119909 119910] sub R+ It is obvious that119891is bounded from above by 119872 = max119891(119909) 119891(119910) sincefor any 119911 = 1199091205821199101minus120582 in the interval 119891(119911) le 119891(119909)120582119891(119910)1minus120582 for
8 International Journal of Analysis
120582 isin [1 119890] It is also bounded from below as we see by writingan arbitrary point in the form 119905radic119909119910 for 119905 isin R+ Then
1198912 (radic119909119910) le 119891 (119905radic119909119910)119891(radic119909119910119905 ) (45)
Using 119872 as the upper bound 119891(radic119909119910119905) we obtain119891 (119905radic119909119910) ge 11198721198912 (radic119909119910) = 119898 (46)
Thus a bigeometric convex functionmay not be continuous atthe boundary points of its domainWewill prove that for anyclosed subinterval [119909 119910] of the interior of the domain thereis a constant 119870 gt 0 so that for any two points 119886 119887 isin [119909 119910] subR+
119891 (119886)119891 (119887) le (119886119887)119870 (47)
A function that satisfies (47) for some 119870 and all 119886 and 119887 inan interval is said to satisfy bigeometric Lipschitz conditionon the interval
Theorem 19 Suppose that 119891 119868 rarr R+ is multiplicative con-vex Then 119891 satisfies the multiplicative Lipschitz condition onany closed interval [119909 119910] sub R+ contained in the interior 1198680 of119868 that is 119891 is continuous on 1198680Proof Take 120576 gt 1 so that [119909120576 119910120576] isin 119868 and let 119898 and 119872 bethe lower and upper bounds for 119891 on [119909120576 119910120576] If 119903 and 119904 aredistinct points of [119909 119910] with 119904 gt 119903 set
119911 = 119904120576120582 = ( 119904119903)
1 ln(120576119904119903) (120582 isin (1 119890)) (48)
Then 119911 isin [119909120576 119910120576] and 119904 = 119911ln1205821199031minusln120582 and we obtain
119891 (119904) le 119891 (119911)ln120582 119891 (119903)1minusln120582 = (119891 (119911)119891 (119903) )ln120582 119891 (119903) (49)
which yields
ln(119891 (119904)119891 (119903)) le ln 120582 ln(119891 (119911)119891 (119903) )lt ln( 119904119903)
1 ln(120576)ln(119872119898 )
119891 (119904)119891 (119903) le ( 119904119903)ln(119872119898) ln(120576)
(50)
where 119870 = ln(119872119898) ln(120576) gt 0 Since the points 119903 119904 isin [119909 119910]are arbitrary we get 119891 that satisfies a multiplicative Lipschitzcondition The remaining part can be obtained in the similarway by taking 119904 lt 119903 and 119911 = 119904120576 Finally 119891 is continuoussince [119909 119910] is arbitrary in 1198680
6 Concluding Remarks
Although all arithmetics are isomorphic only by distinguish-ing among them do we obtain suitable tools for construct-ing all the non-Newtonian calculi But the usefulness ofarithmetic is not limited to the construction of calculi webelieve there is a more fundamental reason for consideringalternative arithmetics theymay also be helpful in developingand understanding new systems of measurement that couldyield simpler physical laws
In this paper it was shown that due to the choice ofgenerator function 119860119901 119866119901 and 119867119901 means are reduced toordinary arithmetic geometric and harmonic mean respec-tively As shown in Table 1 for increasing values of 119901 119860119901and 119901 means increase especially 119901 rarr infin these meansconverge to the value of max119909119899 Conversely for increasingvalues of 119901 119867119901 and 119901 means decrease especially 119901 rarr infinthese means converge to the value of min119909119899 Additionallywe give some new definitions regarding convex functionswhich are plotted on the non-Newtonian coordinate systemObviously for different generator functions one can obtainsome new geometrical interpretations of convex functionsOur future works will include the most famous HermiteHadamard inequality for the class of lowast-convex functionsCompeting Interests
The authors declare that they have no competing interests
References
[1] C Niculescu and L-E Persson Convex Functions and TheirApplications A Contemporary Approach Springer Berlin Ger-many 2006
[2] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000
[3] R Webster Convexity Oxford University Press New York NYUSA 1995
[4] J Banas and A Ben Amar ldquoMeasures of noncompactness inlocally convex spaces and fixed point theory for the sum oftwo operators on unbounded convex setsrdquo CommentationesMathematicae Universitatis Carolinae vol 54 no 1 pp 21ndash402013
[5] J Matkowski ldquoGeneralized weighted and quasi-arithmeticmeansrdquo Aequationes Mathematicae vol 79 no 3 pp 203ndash2122010
[6] D Głazowska and J Matkowski ldquoAn invariance of geometricmean with respect to Lagrangian meansrdquo Journal of Mathemat-ical Analysis and Applications vol 331 no 2 pp 1187ndash1199 2007
[7] J Matkowski ldquoGeneralized weighted quasi-arithmetic meansand the Kolmogorov-Nagumo theoremrdquo Colloquium Mathe-maticum vol 133 no 1 pp 35ndash49 2013
[8] N Merentes and K Nikodem ldquoRemarks on strongly convexfunctionsrdquo Aequationes Mathematicae vol 80 no 1-2 pp 193ndash199 2010
[9] M Grossman Bigeometric Calculus Archimedes FoundationBox 240 Rockport Mass USA 1983
[10] M Grossman and R Katz Non-Newtonian Calculus NewtonInstitute 1978
International Journal of Analysis 9
[11] M Grossman The First Nonlinear System of Differential andIntegral Calculus 1979
[12] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008
[13] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007
[14] E Mısırlı and Y Gurefe ldquoMultiplicative adams-bashforth-moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011
[15] A F Cakmak and F Basar ldquoSome new results on sequencespaces with respect to non-Newtonian calculusrdquo Journal ofInequalities and Applications vol 2012 article 228 2012
[16] A F Cakmak and F Basar ldquoCertain spaces of functions overthe field of non-Newtonian complex numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 236124 12 pages 2014
[17] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013
[18] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014
[19] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015
[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014
[21] U Kadak ldquoNon-Newtonian fuzzy numbers and related applica-tionsrdquo Iranian Journal of Fuzzy Systems vol 12 no 5 pp 117ndash1372015
[22] Y Gurefe U Kadak E Misirli and A Kurdi ldquoA new look atthe classical sequence spaces by using multiplicative calculusrdquoUniversity Politehnica of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 78 no 2 pp 9ndash20 2016
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Journal of Analysis
120582 isin [1 119890] It is also bounded from below as we see by writingan arbitrary point in the form 119905radic119909119910 for 119905 isin R+ Then
1198912 (radic119909119910) le 119891 (119905radic119909119910)119891(radic119909119910119905 ) (45)
Using 119872 as the upper bound 119891(radic119909119910119905) we obtain119891 (119905radic119909119910) ge 11198721198912 (radic119909119910) = 119898 (46)
Thus a bigeometric convex functionmay not be continuous atthe boundary points of its domainWewill prove that for anyclosed subinterval [119909 119910] of the interior of the domain thereis a constant 119870 gt 0 so that for any two points 119886 119887 isin [119909 119910] subR+
119891 (119886)119891 (119887) le (119886119887)119870 (47)
A function that satisfies (47) for some 119870 and all 119886 and 119887 inan interval is said to satisfy bigeometric Lipschitz conditionon the interval
Theorem 19 Suppose that 119891 119868 rarr R+ is multiplicative con-vex Then 119891 satisfies the multiplicative Lipschitz condition onany closed interval [119909 119910] sub R+ contained in the interior 1198680 of119868 that is 119891 is continuous on 1198680Proof Take 120576 gt 1 so that [119909120576 119910120576] isin 119868 and let 119898 and 119872 bethe lower and upper bounds for 119891 on [119909120576 119910120576] If 119903 and 119904 aredistinct points of [119909 119910] with 119904 gt 119903 set
119911 = 119904120576120582 = ( 119904119903)
1 ln(120576119904119903) (120582 isin (1 119890)) (48)
Then 119911 isin [119909120576 119910120576] and 119904 = 119911ln1205821199031minusln120582 and we obtain
119891 (119904) le 119891 (119911)ln120582 119891 (119903)1minusln120582 = (119891 (119911)119891 (119903) )ln120582 119891 (119903) (49)
which yields
ln(119891 (119904)119891 (119903)) le ln 120582 ln(119891 (119911)119891 (119903) )lt ln( 119904119903)
1 ln(120576)ln(119872119898 )
119891 (119904)119891 (119903) le ( 119904119903)ln(119872119898) ln(120576)
(50)
where 119870 = ln(119872119898) ln(120576) gt 0 Since the points 119903 119904 isin [119909 119910]are arbitrary we get 119891 that satisfies a multiplicative Lipschitzcondition The remaining part can be obtained in the similarway by taking 119904 lt 119903 and 119911 = 119904120576 Finally 119891 is continuoussince [119909 119910] is arbitrary in 1198680
6 Concluding Remarks
Although all arithmetics are isomorphic only by distinguish-ing among them do we obtain suitable tools for construct-ing all the non-Newtonian calculi But the usefulness ofarithmetic is not limited to the construction of calculi webelieve there is a more fundamental reason for consideringalternative arithmetics theymay also be helpful in developingand understanding new systems of measurement that couldyield simpler physical laws
In this paper it was shown that due to the choice ofgenerator function 119860119901 119866119901 and 119867119901 means are reduced toordinary arithmetic geometric and harmonic mean respec-tively As shown in Table 1 for increasing values of 119901 119860119901and 119901 means increase especially 119901 rarr infin these meansconverge to the value of max119909119899 Conversely for increasingvalues of 119901 119867119901 and 119901 means decrease especially 119901 rarr infinthese means converge to the value of min119909119899 Additionallywe give some new definitions regarding convex functionswhich are plotted on the non-Newtonian coordinate systemObviously for different generator functions one can obtainsome new geometrical interpretations of convex functionsOur future works will include the most famous HermiteHadamard inequality for the class of lowast-convex functionsCompeting Interests
The authors declare that they have no competing interests
References
[1] C Niculescu and L-E Persson Convex Functions and TheirApplications A Contemporary Approach Springer Berlin Ger-many 2006
[2] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000
[3] R Webster Convexity Oxford University Press New York NYUSA 1995
[4] J Banas and A Ben Amar ldquoMeasures of noncompactness inlocally convex spaces and fixed point theory for the sum oftwo operators on unbounded convex setsrdquo CommentationesMathematicae Universitatis Carolinae vol 54 no 1 pp 21ndash402013
[5] J Matkowski ldquoGeneralized weighted and quasi-arithmeticmeansrdquo Aequationes Mathematicae vol 79 no 3 pp 203ndash2122010
[6] D Głazowska and J Matkowski ldquoAn invariance of geometricmean with respect to Lagrangian meansrdquo Journal of Mathemat-ical Analysis and Applications vol 331 no 2 pp 1187ndash1199 2007
[7] J Matkowski ldquoGeneralized weighted quasi-arithmetic meansand the Kolmogorov-Nagumo theoremrdquo Colloquium Mathe-maticum vol 133 no 1 pp 35ndash49 2013
[8] N Merentes and K Nikodem ldquoRemarks on strongly convexfunctionsrdquo Aequationes Mathematicae vol 80 no 1-2 pp 193ndash199 2010
[9] M Grossman Bigeometric Calculus Archimedes FoundationBox 240 Rockport Mass USA 1983
[10] M Grossman and R Katz Non-Newtonian Calculus NewtonInstitute 1978
International Journal of Analysis 9
[11] M Grossman The First Nonlinear System of Differential andIntegral Calculus 1979
[12] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008
[13] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007
[14] E Mısırlı and Y Gurefe ldquoMultiplicative adams-bashforth-moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011
[15] A F Cakmak and F Basar ldquoSome new results on sequencespaces with respect to non-Newtonian calculusrdquo Journal ofInequalities and Applications vol 2012 article 228 2012
[16] A F Cakmak and F Basar ldquoCertain spaces of functions overthe field of non-Newtonian complex numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 236124 12 pages 2014
[17] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013
[18] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014
[19] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015
[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014
[21] U Kadak ldquoNon-Newtonian fuzzy numbers and related applica-tionsrdquo Iranian Journal of Fuzzy Systems vol 12 no 5 pp 117ndash1372015
[22] Y Gurefe U Kadak E Misirli and A Kurdi ldquoA new look atthe classical sequence spaces by using multiplicative calculusrdquoUniversity Politehnica of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 78 no 2 pp 9ndash20 2016
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 9
[11] M Grossman The First Nonlinear System of Differential andIntegral Calculus 1979
[12] A E Bashirov E M Kurpınar and A Ozyapıcı ldquoMultiplicativecalculus and its applicationsrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 36ndash48 2008
[13] D Aniszewska ldquoMultiplicative Runge-Kutta methodsrdquo Nonlin-ear Dynamics vol 50 no 1-2 pp 265ndash272 2007
[14] E Mısırlı and Y Gurefe ldquoMultiplicative adams-bashforth-moulton methodsrdquo Numerical Algorithms vol 57 no 4 pp425ndash439 2011
[15] A F Cakmak and F Basar ldquoSome new results on sequencespaces with respect to non-Newtonian calculusrdquo Journal ofInequalities and Applications vol 2012 article 228 2012
[16] A F Cakmak and F Basar ldquoCertain spaces of functions overthe field of non-Newtonian complex numbersrdquo Abstract andApplied Analysis vol 2014 Article ID 236124 12 pages 2014
[17] S Tekin and F Basar ldquoCertain sequence spaces over the non-Newtonian complex fieldrdquo Abstract and Applied Analysis vol2013 Article ID 739319 11 pages 2013
[18] U Kadak and H Efe ldquoMatrix transformations between certainsequence spaces over the non-Newtonian complex fieldrdquo TheScientific World Journal vol 2014 Article ID 705818 12 pages2014
[19] U Kadak and M Ozluk ldquoGeneralized Runge-Kutta methodwith respect to the non-Newtonian calculusrdquo Abstract andApplied Analysis vol 2015 Article ID 594685 10 pages 2015
[20] U Kadak ldquoDetermination of the Kothe-Toeplitz duals over thenon-Newtonian complex fieldrdquoTheScientificWorld Journal vol2014 Article ID 438924 10 pages 2014
[21] U Kadak ldquoNon-Newtonian fuzzy numbers and related applica-tionsrdquo Iranian Journal of Fuzzy Systems vol 12 no 5 pp 117ndash1372015
[22] Y Gurefe U Kadak E Misirli and A Kurdi ldquoA new look atthe classical sequence spaces by using multiplicative calculusrdquoUniversity Politehnica of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 78 no 2 pp 9ndash20 2016
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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