Research Article Second Hankel Determinants for the Class of Typically Real Functions PaweB Zaprawa Department of Mathematics, Lublin University of Technology, Nadbystrzycka 38D, 20-618 Lublin, Poland Correspondence should be addressed to Paweł Zaprawa; [email protected] Received 5 November 2015; Revised 22 December 2015; Accepted 3 January 2016 Academic Editor: Marco Donatelli Copyright © 2016 Paweł Zaprawa. 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. We discuss the Hankel determinants 2 () = +2 − ( +1 ) 2 for typically real functions, that is, analytic functions which satisfy the condition ImIm() ≥ 0 in the unit disk Δ. Main results are concerned with 2 (2) and 2 (3). e sharp upper and lower bounds are given. In general case, for ≥4, the results are not sharp. Moreover, we present some remarks connected with typically real odd functions. 1. Introduction Let Δ be the unit disk { ∈ C : || < 1} and let A be the family of all functions analytic in Δ that have the Taylor series expansion () = + ∑ ∞ =2 . In [1, 2] Pommerenke defined th Hankel determinant for a function as () = +1 ⋅⋅⋅ +−1 +1 +2 ⋅⋅⋅ + ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ +−1 + ⋅⋅⋅ +2−2 , (1) where , ∈ N. Recently, the Hankel determinant has been studied inten- sively by many mathematicians. e research was focused on 2 (2) for various classes of univalent functions. e papers by Janteng et al. [3, 4], Lee et al. [5], Vamshee Krishna and Ramreddy [6], and Selvaraj and Kumar [7] are worth mentioning here. Janteng et al. derived the exact bounds of | 2 (2)| for the classes: ∗ of star-like functions (| 2 (2)| ≤ 1), K of convex functions (| 2 (2)| ≤ 1/8), and R of functions whose derivative has a positive real part (| 2 (2)| ≤ 4/9). Lee et al. [5] investigated the Hankel determinant in the general class ∗ () of star-like functions with respect to a given function . is class was defined by Ma and Minda in [8]. In particular, Lee et al. obtained the results for the following classes: ∗ () of star-like functions of order (| 2 (2)| ≤ (1 − ) 2 ), SL ∗ of lemniscate star-like functions (| 2 (2)| ≤ 1/16; for the definition of SL ∗ , see [9]), and ∗ of strongly star- like functions of order (| 2 (2)| ≤ 2 ). Vamshee Krishna and Ramreddy [6] generalized the result of Janteng et al. ey gave the bound of | 2 (2)| in the class K() of convex functions of order . Selvaraj and Kumar [7] proved that the estimate of the second Hankel determinant for the class C of close-to-convex functions is the same as that for the class ∗ . e question whether this bound is good for the class of all univalent functions has no answer yet. One can find some other results in this direction in [10–14]. Taking different set of parameters and , the research on the Hankel determinant is much more difficult. In [15] Hayami and Owa discussed 2 () for functions satisfying Re()/ > or Re () > . On the other hand, Babalola [16] tried to estimate | 3 (1)| for ∗ , K, and R. Shanmugam et al. [17] discussed | 3 (1)| for the class of -star-like functions defined by Mocanu in [18]. In particular, if =2 and =1 then 2 (1) is known as a classical functional of Fekete-Szeg¨ o. A lot of papers have been devoted to the studies concerning this functional. Because 2 (1) is not related to the subject of this paper, we omit recalling results obtained in this direction. e majority of results concerning the Hankel determi- nants were obtained for univalent functions. In this paper we discuss functions which, in general, are not univalent. We focus our investigation on typically real functions. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2016, Article ID 3792367, 7 pages http://dx.doi.org/10.1155/2016/3792367
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Research ArticleSecond Hankel Determinants for the Class ofTypically Real Functions
PaweB Zaprawa
Department of Mathematics Lublin University of Technology Nadbystrzycka 38D 20-618 Lublin Poland
Correspondence should be addressed to Paweł Zaprawa pzaprawapollubpl
Received 5 November 2015 Revised 22 December 2015 Accepted 3 January 2016
Academic Editor Marco Donatelli
Copyright copy 2016 Paweł Zaprawa This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We discuss the Hankel determinants 1198672(119899) = 119886
119899119886119899+2
minus (119886119899+1
)2 for typically real functions that is analytic functions which satisfy
the condition Im119911Im119891(119911) ge 0 in the unit disk Δ Main results are concerned with 1198672(2) and 119867
2(3) The sharp upper and lower
bounds are given In general case for 119899 ge 4 the results are not sharp Moreover we present some remarks connected with typicallyreal odd functions
1 Introduction
Let Δ be the unit disk 119911 isin C |119911| lt 1 and let A be thefamily of all functions 119891 analytic in Δ that have the Taylorseries expansion 119891(119911) = 119911 + sum
infin
119899=2119886119899119911119899 In [1 2] Pommerenke
defined 119902th Hankel determinant for a function 119891 as
119867119902(119899) =
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119886119899
119886119899+1
sdot sdot sdot 119886119899+119902minus1
119886119899+1
119886119899+2
sdot sdot sdot 119886119899+119902
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
119886119899+119902minus1
119886119899+119902
sdot sdot sdot 119886119899+2119902minus2
100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(1)
where 119899 119902 isin NRecently the Hankel determinant has been studied inten-
sively by many mathematiciansThe research was focused on1198672(2) for various classes of univalent functions The papers
by Janteng et al [3 4] Lee et al [5] Vamshee Krishnaand Ramreddy [6] and Selvaraj and Kumar [7] are worthmentioning here Janteng et al derived the exact bounds of|1198672(2)| for the classes 119878lowast of star-like functions (|119867
2(2)| le 1)
K of convex functions (|1198672(2)| le 18) and R of functions
whose derivative has a positive real part (|1198672(2)| le 49) Lee
et al [5] investigated the Hankel determinant in the generalclass 119878
lowast(120593) of star-like functions with respect to a given
function 120593 This class was defined by Ma and Minda in [8]In particular Lee et al obtained the results for the followingclasses 119878lowast(120572) of star-like functions of order 120572 (|119867
2(2)| le (1 minus
120572)2) SLlowast of lemniscate star-like functions (|119867
2(2)| le 116
for the definition of SLlowast see [9]) and 119878lowast
120573of strongly star-
like functions of order 120573 (|1198672(2)| le 120573
2) Vamshee Krishnaand Ramreddy [6] generalized the result of Janteng et alThey gave the bound of |119867
2(2)| in the class K(120572) of convex
functions of order 120572 Selvaraj and Kumar [7] proved that theestimate of the second Hankel determinant for the classC ofclose-to-convex functions is the same as that for the class 119878lowastThe question whether this bound is good for the class 119878 ofall univalent functions has no answer yet One can find someother results in this direction in [10ndash14]
Taking different set of parameters 119902 and 119899 the researchon the Hankel determinant is much more difficult In [15]Hayami and Owa discussed 119867
2(119899) for functions 119891 satisfying
Re119891(119911)119911 gt 120572 or Re1198911015840(119911) gt 120572 On the other hand Babalola[16] tried to estimate |119867
3(1)| for 119878lowastK andR Shanmugam
et al [17] discussed |1198673(1)| for the class 119872
120572of 120572-star-like
functions defined by Mocanu in [18]In particular if 119902 = 2 and 119899 = 1 then119867
2(1) is known as a
classical functional of Fekete-Szego A lot of papers have beendevoted to the studies concerning this functional Because1198672(1) is not related to the subject of this paper we omit
recalling results obtained in this directionThe majority of results concerning the Hankel determi-
nants were obtained for univalent functions In this paperwe discuss functions which in general are not univalent Wefocus our investigation on typically real functions
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2016 Article ID 3792367 7 pageshttpdxdoiorg10115520163792367
2 Abstract and Applied Analysis
2 Class 119879 and the Hankel Determinants fora Selected Functions in 119879
A function 119891 isin A that satisfies the condition Im 119911 Im119891(119911) ge
0 for 119911 isin Δ is called a typically real function Let 119879 denotethe class of all typically real functions Robertson [19] provedthat 119891 isin 119879 if and only if there exists a probability measure 120583on [minus1 1] such that the following formula holds
119891 (119911) = int
1
minus1
119911
1 minus 2119911119905 + 1199112119889120583 (119905) (2)
The coefficients of a function 119891(119911) = 119911 + suminfin
119899=2119886119899119911119899isin 119879 can
be written as follows
119886119899= int
1
minus1
sin (119899 arccos 119905)sin (arccos 119905)
119889120583 (119905) = int
1
minus1
119880119899minus1
(119905) 119889120583 (119905)
119899 = 1 2
(3)
The functions 119880119899(119905) 119899 = 1 2 which appear in the above
formula are the well-known Chebyshev polynomials of thesecond kind
Since all coefficients of 119891 isin 119879 are real we look for thelower and the upper bounds of119867
2(2) instead of the bound of
|1198672(2)| At the beginning let us look at a few examples
Example 1 All the functions 119891119905(119911) = 119911(1 minus 2119911119905 + 119911
2) 119905 isin
[minus1 1] are in 119878lowast Since 119891119905(119911) = 119911 + 2119905119911
2+ (41199052minus 1)1199113+ (81199053minus
4119905)1199114+sdot sdot sdot we have119867
2(2) = minus1 for each 119905 isin [minus1 1]Moreover
1198672(119899) = 119880
119899minus1(119905)119880119899+1
(119905) minus 119880119899(119905)2 This and the Turan identity
for Chebyshev polynomials 119880119899(119905) result in 119867
2(119899) = minus1 for
each 119899 = 2 3
Example 2 For a function 119891(119911) = 119911(1 + 1199112)(1 minus 119911
2)2 having
the Taylor series expansion 119891(119911) = 119911 + 31199113+ 51199115+ sdot sdot sdot there
is1198672(119899) = minus(119899 + 1)
2 for even 119899 and1198672(119899) = (119899 + 1)
2minus 1 for
odd 119899 In this case the function 119891 is not univalent the boundof |1198672(2)| is much greater than 1 the value of the second
Hankel determinant for star-like functions or close-to-convexfunctions
Example 3 EveryHankel determinant1198672(119899) 119899 = 1 2 for
a function 119891(119911) = log(1(1 minus 119911)) = 119911+ (12)1199112+ (13)119911
3+ sdot sdot sdot
is positive Namely1198672(119899) = 1119899(119899 + 1)
2(119899 + 2)
For a given class 119860 sub A we denote by Ω119899(119860) 119899 ge 1
the region of variability of three succeeding coefficients offunctions in 119860 that is the set (119886
119899(119891) 119886119899+1
(119891) 119886119899+2
(119891))
119891 isin 119860 As it is seen in (3) the coefficients of typicallyreal functions are the Stieltjes integrals of the Chebyshevpolynomials of the second kind with respect to a probabilitymeasure HenceΩ
119899(119879) is the closed convex hull of the curve
120574 [minus1 1] ni 119905 rarr (119880119899minus1
(119905) 119880119899(119905) 119880119899+1
(119905)) (see eg [20])
Lemma 4 The functional 119879 ni 119891 rarr 1198672(119899) 119899 ge 2 attains its
extreme values on the boundary of Ω119899(119879)
Proof The only critical point of ℎ(119909 119910 119911) = 119909119911 minus 1199102 where
119909 = 119886119899 119910 = 119886
119899+1 and 119911 = 119886
119899+2 is (0 0 0) But ℎ(0 0 0) = 0
Since ℎ may be positive as well as negative for (119909 119910 119911) isin
Ω119899(119879) (see Examples 1 and 3) it means that the extreme
values of ℎ are attained on the boundary ofΩ119899(119879)
3 Bounds of 1198672(2) in 119879
In [21] Ma proved so-called generalized Zalcman conjecturefor the class 119879
1003816100381610038161003816119886119899119886119898 minus 119886119899+119898minus1
1003816100381610038161003816
le
119899 + 1 119898 = 2 119899 = 2 4 6
119898 + 1 119899 = 2 119898 = 2 4 6
(119899 minus 1) (119898 minus 1) otherwise
(4)
We apply this result to prove the following
Theorem 5 If 119891 isin 119879 then |1198672(2)| le 9
Proof The result of Ma and the triangle inequality result in1003816100381610038161003816100381611988621198864minus 1198863
210038161003816100381610038161003816le100381610038161003816100381611988621198864 minus 119886
5
1003816100381610038161003816 +100381610038161003816100381610038161198865minus 1198863
210038161003816100381610038161003816le 5 + 4 = 9 (5)
This result is sharp the equality holds for 119891(119911) = 119911(1 +
1199112)(1 minus 119911
2)2 Furthermore we can see the following
Corollary 6 For 119879 one has
min 1198672(2) 119891 isin 119879 = minus9 (6)
For our next theorem let us cite two results First one isthe obvious conclusion from the Caratheodory theorem andthe Krein-Milman theorem We assume that 119883 is a compactHausdorff space and
119869120583= int119883
119869 (119905) 119889120583 (119905) (7)
Theorem A (see [22 Thm 140]) If 119869 119883 rarr R119899 is continu-ous then the convex hull of 119869(119883) is a compact set and it coincideswith the set 119869
120583 120583 isin 119875
119883 |supp(120583)| le 119899
In the above the symbols 119875119883and |supp(120583)| stand for the
set of probability measures on 119883 and the cardinality of thesupport of 120583 respectively
It means that 120583 is atomic measure having at most 119899 stepsMore precise information about the relation between themeasure and the convex hull is presented in the followingtheorem In what follows ⟨ 119886 ⟩ means the scalar product of119886 and
Theorem B (see [22 Thm 149]) Let 119869 [120572 120573] rarr R119899 becontinuous Suppose that there exists a positive integer 119896 suchthat for each nonzero 119901 in R119899 the number of solutions of anyequation ⟨
997888997888rarr119869(119905) ⟩ = const 120572 le 119905 le 120573 is not greater than 119896
Abstract and Applied Analysis 3
Then for every 120583 isin 119875[120572120573]
such that 119869120583belongs to the boundary
of the convex hull of 119869([120572 120573]) the following statements aretrue
(1) If 119896 = 2119898 then
(a) |supp(120583)| le 119898 or(b) |supp(120583)| = 119898 + 1 and 120572 120573 sub supp(120583)
(2) If 119896 = 2119898 + 1 then
(a) |supp(120583)| le 119898 or(b) |supp(120583)| = 119898 + 1 and one of the points 120572 and 120573
belongs to supp(120583)
This theorem in slightly modified version was publishedin [23] as Lemma 2
Putting 119869(119905) = [1198801(119905) 1198802(119905) 1198803(119905)] 119905 isin [minus1 1] and =
[1199011 1199012 1199013] we can see that any equation of the form
11990111198801(119905) + 119901
21198802(119905) + 119901
31198803(119905) = const 119905 isin [minus1 1] (8)
is equivalent to 1198823(119905) = const where 119882
3(119905) is a polynomial
of degree 3 Hence (8) has at most 3 solutions According toTheorem B the boundary of the convex hull of 119869([minus1 1]) isdetermined by atomic measures 120583 for which support consistsof at most 2 points Moreover one of them has to be minus1 or 1We have proved the following
Lemma7 Theboundary ofΩ2(119879) consists of points (119886
2 1198863 1198864)
that correspond to the following functions
119891 (119911) = 120572119911
1 minus 2119911119905 + 1199112+ (1 minus 120572)
119911
(1 minus 119911)2
120572 isin [0 1] 119905 isin [minus1 1]
(9)
or
119891 (119911) = 120572119911
1 minus 2119911119905 + 1199112+ (1 minus 120572)
119911
(1 + 119911)2
120572 isin [0 1] 119905 isin [minus1 1]
(10)
Now we are ready to prove the following
Theorem 8 For 119879 one has
max 1198672(2) 119891 isin 119879 = 1 (11)
Proof By Lemma 7 it is enough to take functions given by(9) or (10) Consider the following
(I) Function (9) has the series expansion
119891 (119911) = 119911 + 2 (1 minus 120572 + 120572119905) 1199112
+ [3 (1 minus 120572) + (41199052minus 1) 120572] 119911
3
+ 4 [1 minus 120572 + (21199053minus 119905) 120572] 119911
4+ sdot sdot sdot
(12)
Hence1198672(2) = 119892
1(120572 119905) where
1198921(120572 119905) = 8 (1 minus 120572 + 120572119905) [1 minus 120572 + (2119905
3minus 119905) 120572]
minus (41199052120572 minus 4120572 + 3)
2
120572 isin [0 1] 119905 isin [minus1 1]
(13)
From
1205971198921
120597120572= 8 (1 minus 119905
2) (2 (2 minus 120572) 119905
2+ 1 minus 2120572)
1205971198921
120597119905= 16120572 (120572 minus 1) (4119905
2minus 3119905 + 3)
(14)
it follows that the critical points of 1198921are as follows (0 minus1)
(0 1) (1 minus1) (1 1) (1 minus1radic2) (1 1radic2) and (12 0)Among these points only (12 0) lies inside the set [0 1] times[minus1 1]
If 120572 = 0 or 120572 = 1 then functions (9) coincide with 119891119905
from Example 1 If 119905 = 1 then 119891(119911) = 119911(1 minus 119911)2 In each
case 1198672(2) = minus1 For 119905 = minus1 function (9) takes the form
119891(119911) = 120572(119911(1 + 119911)2) + (1 minus 120572)(119911(1 minus 119911)
2) Then 119867
2(2) =
8(1 minus 2120572)2minus 9 le minus1
If 120572 = 12 and 119905 = 0 we have1198672(2) = 1 It means that the
greatest value of1198672(2) for functions given by (9) is equal to 1
The extremal function is
119891 (119911) =1
2[
119911
1 + 1199112+
119911
(1 minus 119911)2]
= 119911 + 1199112+ 1199113+ 21199114+ 31199115+ sdot sdot sdot
(15)
(II) For functions (10)1198672(2) is equal to 119892
2(120572 119905) where
1198922(120572 119905) = 8 (minus1 + 120572 + 120572119905) [minus1 + 120572 + (2119905
3minus 119905) 120572]
minus (41199052120572 minus 4120572 + 3)
2
120572 isin [0 1] 119905 isin [minus1 1]
(16)
Moreover 1198922(120572 119905) = 119892
1(120572 minus119905) Taking into account the sym-
metry of the range of variability of 119905 we obtain the sameresult as above also for functions defined by (10)The extremalfunction is
119891 (119911) =1
2[
119911
1 + 1199112+
119911
(1 + 119911)2]
= 119911 minus 1199112+ 1199113minus 21199114+ 31199115+ sdot sdot sdot
(17)
4 Bounds of 1198672(3) in 119879
The proof of the following theorem is obvious
Theorem 9 If 119899 is odd then
max 1198672(119899) 119891 isin 119879 = 119899 (119899 + 2) (18)
Hence one has the following
4 Abstract and Applied Analysis
Corollary 10 For 119879 one has
max 1198672(3) 119891 isin 119879 = 15 (19)
In similar way as it was done for Lemma 7 one can provethe following
Lemma 11 The boundary of Ω3(119879) consists of points (119886
3
1198864 1198865) that correspond to the following functions
119891 (119911) = 120572119911
1 minus 21199111199051+ 1199112
+ (1 minus 120572)119911
1 minus 21199111199052+ 1199112
120572 isin [0 1] 1199051 1199052 isin [minus1 1]
(20)
or
119891 (119911) = 120572119911
(1 + 119911)2+ 120573
119911
(1 minus 119911)2
+ (1 minus 120572 minus 120573)119911
1 minus 2119911119905 + 1199112
120572 120573 isin [0 1] 120572 + 120573 le 1 119905 isin [minus1 1]
(21)
Theorem 12 For 119879 one has
min 1198672(3) 119891 isin 119879 = minus
4
3+8
9
radic6 = minus351 (22)
Proof By Lemma 11 it suffices to discuss functions given by(20) or (21) Consider the following
(I) For functions (20) we have
1198863= (41199051
2minus 1) 120572 + (4119905
2
2minus 1) (1 minus 120572)
1198864= (81199051
3minus 41199051) 120572 + (8119905
2
3minus 41199052) (1 minus 120572)
1198865= (16119905
1
4minus 121199051
2+ 1) 120572
+ (161199052
4minus 121199052
2+ 1) (1 minus 120572)
(23)
and hence applying the Turan identity1198672(3) = 119892
3(120572 1199051 1199052)
where
1198923(120572 1199051 1199052)
= minus1205722minus (1 minus 120572)
2
+ 2120572 (1 minus 120572) [8 (1199051minus 1199052)2
(1 minus 1199051
2) (1 minus 119905
2
2) minus 1]
120572 isin [0 1] 1199051 1199052 isin [minus1 1]
(24)
The expression in brackets is greater than or equal to minus1for all 119905
1 1199052isin [minus1 1] Hence
1198672(3) ge minus120572
2minus (1 minus 120572)
2minus 2120572 (1 minus 120572) = minus1 (25)
(II) If function 119891 is given by (21) then
1198863= 3 (120572 + 120573) + (4119905
2minus 1) (1 minus 120572 minus 120573)
1198864= 4 (120573 minus 120572) + (8119905
3minus 4119905) (1 minus 120572 minus 120573)
1198865= 5 (120572 + 120573) + (16119905
4minus 121199052+ 1) (1 minus 120572 minus 120573)
(26)
Using the Turan identity it follows that 1198672(3) = 119892
4(120572 120573 119905)
where
1198924(120572 120573 119905) = minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)
2
+ 64120572120573
+ 2 (1 minus 120572 minus 120573) 119902 (119905)
119902 (119905) = (120572 + 120573) (241199054minus 81199052minus 1)
+ (120572 minus 120573) (321199053minus 16119905)
(27)
under the assumptions 120572 120573 isin [0 1] 120572+120573 le 1 and 119905 isin [minus1 1]Let 120572 and 120573 be fixed Since
120597119902
120597119905= 8 (6119905
2minus 1) [119905 (120572 + 120573) + 120572 minus 120573] (28)
the critical points of 119902 are as follows minus1radic6 1radic6 and (120573 minus
120572)(120572 + 120573) It is easily seen that all these points are in [minus1 1]Therefore
min 119902 (119905) 119905 isin [minus1 1] = min119902 (minus1) 119902 ( minus1
radic6
)
119902 (1
radic6
) 119902 (1) 119902 (120573 minus 120572
120572 + 120573)
(29)
For 119905 = minus1 or 119905 = 1 the functions given by (21) have theform
119891 (119911) = 119911
(1 + 119911)2+ 120573
119911
(1 minus 119911)2 + 120573 = 1 (30)
One can show directly from formula (30) that
1198672(3) = minus1 + 64 (1 minus ) ge minus1 (31)
For 119905 = (120573 minus 120572)(120572 + 120573) there is
1198672(3) = minus1 +
64120572120573 [4 (120572 + 120573) 120572120573 + (120573 minus 120572)2
]
(120572 + 120573)3
(32)
hence
1198672(3) ge minus1 (33)
If 119905 = minus1radic6 or 119905 = 1radic6 then1198672(3) is equal to
1198672(3)
= minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)2
+ 64120572120573
+ (1 minus 120572 minus 120573) [minus10
3(120572 + 120573) minus
64
3radic6
(120572 minus 120573)]
(34)
or
1198672(3)
= minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)2
+ 64120572120573
+ (1 minus 120572 minus 120573) [minus10
3(120572 + 120573) +
64
3radic6
(120572 minus 120573)]
(35)
Abstract and Applied Analysis 5
respectively Without loss of generality we can assume that120572 ge 120573 Then while looking for the minimum value of119867
2(3)
we can restrict the research to the first stated above case (sinceexpression (35) is not less than expression (34))
Transforming (35) we obtain
1198672(3) = minus1 + 64120572120573 minus
4
3(120572 + 120573) (1 minus 120572 minus 120573)
+64
3radic6
(120573 minus 120572) (1 minus 120572 minus 120573)
(36)
Taking the smallest possible 120573 (ie 120573 = 0) the second andthe forth component of this expression will not increase Thevalue of the third component does not depend only on 120573 infact it depends on 120572 + 120573 For this reason we can take 120573 = 0Combining these facts it yields that
1198672(3) ge minus1 minus
4
3120572 (1 minus 120572) minus
64
3radic6
120572 (1 minus 120572) (37)
The smallest value of the right hand side of this inequality isachieved for 120572 = 12 In this case
1198672(3) ge minus1 minus
1
3minus
16
3radic6
= minus4
3minus8
9
radic6 = minus351 (38)
Combining two parts of the proof we obtain the conclu-sion of the theorem Furthermore the above shows that theextremal functions are
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 21199111199050+ 1199112
]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 + 21199111199050+ 1199112
]
(39)
where 1199050= 1radic6
5 Bounds of 1198672(119899) 119899 ge 4 in 119879
It is easily seen that 1198672(119899) le 119899(119899 + 2) for any typically real
function ByTheorem 9 this estimate is sharp providing that119899 is an odd integer At the beginning of this section we willprove the following
Theorem 13 For 119879 one has
min 119886119899119886119899+2
119891 isin 119879 = minus1 (40)
Proof Thecoefficients of the series expansion of function119891 isin
119879 can be written as follows
119886119899= int
120587
0
sin (119899120579)sin 120579
119889] (120579) ] isin 119875[0120587]
(41)
Hence
119886119899119886119899+2
= int
120587
0
sin ((119899 + 1) 120579 minus 120579)
sin 120579119889] (120579)
sdot int
120587
0
sin ((119899 + 1) 120579 + 120579)
sin 120579119889] (120579)
= (int
120587
0
sin (119899 + 1) 120579
sin 120579cos 120579 119889] (120579))
2
minus (int
120587
0
cos (119899 + 1) 120579 119889] (120579))2
(42)
Since
int
120587
0
cos (119899 + 1) 120579 119889] (120579) le int
120587
0
119889] (120579) = 1 (43)
we obtain
119886119899119886119899+2
ge minus1 (44)
In order to prove that the estimate is sharp let us take themeasure ] for which support satisfies condition (119899 + 1)120579 =
120587 This measure corresponds to the function 119891(119911) = 119911(1 minus
2119911cos(120587(119899 + 1)) + 1199112)
Observe that 119886119899119886119899+2
= minus1 holds not only for the measurestated above Namely the value minus1 in (42) is taken also if (119899 +1)120579 = 119896120587 where 119896 is any positive integer less than or equal to119899 From this we conclude that the support of the measure has119899 points 120579
119896= 119896120587(119899 + 1) with weights 120572
119896 119896 = 1 2 119899 such
that sum119899119896=1
120572119896= 1
The weights satisfy
(
119899
sum
119896=1
120572119896(minus1)119896)
2
= 1 (45)
Indeed if the support of ] consists of 119899 points then 119891 takesthe form
119891 (119911) =
119899
sum
119896=1
120572119896
119911
1 minus 2119911 cos 120579119896+ 1199112
(46)
Using trigonometric identities we obtain
119886119899=
119899
sum
119896=1
120572119896(minus1)119896+1
119886119899+2
=
119899
sum
119896=1
120572119896(minus1)119896
(47)
which results in (45)Connecting (45) andsum119899
119896=1120572119896= 1we conclude that119891 is of
the form
119891 (119911) =
119899
sum
119896=1119896 is odd120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112(48)
6 Abstract and Applied Analysis
Table 1 The bounds of the Hankel determinants for functions defined by (51)
Bounds of1198672(119899) Equality in the lower bound for Equality in the upper bound for
minus9 le 1198672(2) le 1 119905 = 1 119905 = 0
minus351 le 1198672(3) le 15 119905 = 0 40 119905 = 1
minus25 le 1198672(4) le 446 119905 = 1 119905 = 061
minus784 le 1198672(5) le 35 119905 = 0 72 119905 = 1
minus49 le 1198672(6) le 967 119905 = 1 119905 = 079
or
119891 (119911) =
119899
sum
119896=1119896 is even120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112 (49)
Itmeans that for even 119899 the support of ] consists of 1198992 pointsand for even 119899 the number of points of the support of ] is equalto (119899 + 1)2 or (119899 minus 1)2
Taking into account |119886119899+1
| le 119899 + 1 and Theorem 13 weobtain the following
Theorem 14 For 119879 one has
1198672(119899) ge minus (119899 + 1)
2minus 1 (50)
Unfortunately this bound is not sharp However thefollowing can be conjectured
Conjecture 15 For any positive integer 119899 the following esti-mate 119867
2(119899) ge minus(119899 + 1)
2 holds Moreover this bound is sharpfor even 119899
This conjecture is supported by the facts that in thetheorems concerning119867
2(2) and119867
2(3) the extremal functions
are of the form
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 2119911119905 + 1199112]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 minus 2119911119905 + 1199112]
(51)
for appropriately taken 119905 isin [minus1 1] The exact bounds ofthe Hankel determinants for these functions are collected inTable 1 They were obtained numerically
6 Remarks Concerning 1198672(119899) in 119879
(2)
In class 119879 we discuss subclass 119879(2) consisting of the functionswhich are odd The definition of this class is
119879(2)
= 119891 isin 119879 119891 (minus119911) = minus119891 (119911) 119911 isin Δ (52)
For 119891 isin 119879(2) the representation formula similar to (2) is
valid Namely
119891 (119911) = int
1
minus1
119911 (1 + 1199112)
(1 + 1199112)2
minus 411991121199052119889] (119905) ] isin 119875
[minus11] (53)
Function 119891 has the Taylor series expansion
119891 (119911) = sum
119899 is odd119886119899119911119899 119886119899= int
1
minus1
119880119899minus1
(119905) 119889] (119905) (54)
The following inequalities are obvious
minus (119899 + 1)2le 1198672(119899) le 0 for even 119899
1198672(119899) le (119899 + 1)
2minus 1 for odd 119899
(55)
equalities hold for 119891(119911) = 119911(1 + 1199112)(1 minus 119911
2)2
For a given class 119860 sub A let us denote by Ψ119899(119860) 119899 ge 1
the set (119886119899 119886119899+2
) 119891 isin 119860 From (53) it follows that Ψ119899(119879) is
the closed convex hull of the curve
120582 [minus1 1] ni 119905 997888rarr (119880119899minus1
(119905) 119880119899+1
(119905)) (56)
FromTheorem 13 and from the equivalence
(119886119899 119886119899+2
) isin Ψ119899(119879) lArrrArr (119886
119899 119886119899+2
) isin Ψ119899(119879(2)) (57)
we get
min 119886119899119886119899+2
119891 isin 119879(2) = minus1 (58)
Hence for odd 119899 we know that
min 1198672(119899) 119891 isin 119879
(2) ge minus1 (59)
The equality holds for functions (48) or (49) providing that120572119896
= 120572119899+1minus119896
Then connecting the components of theseformulae in pairs we obtain
120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112
+ 120572119899+1minus119896
119911
1 minus 2119911 cos ((119899 + 1 minus 119896) 120587 (119899 + 1)) + 1199112
= 2120572119896
119911 (1 + 1199112)
(1 + 1199112)2
minus 41199112cos2 (119896120587 (119899 + 1))
(60)
With help of the argument given in the proof of Theorem 13we eventually obtain the odd functions for which 119886
119899119886119899+2
=
minus1
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 7
References
[1] C Pommerenke ldquoOn the coefficients and Hankel determinantsof univalent functionsrdquo Journal of the London MathematicalSociety vol 41 pp 111ndash122 1966
[2] C Pommerenke ldquoOn the Hankel determinants of univalentfunctionsrdquoMathematika vol 14 pp 108ndash112 1967
[3] A Janteng S A Halim and M Darus ldquoCoefficient inequalityfor a function whose derivative has a positive real partrdquo Journalof Inequalities in Pure and Applied Mathematics vol 7 no 2article 50 2006
[4] A Janteng S A Halim andMDarus ldquoHankel determinant forstarlike and convex functionsrdquo International Journal of Mathe-matical Analysis vol 1 no 13 pp 619ndash625 2007
[5] S K Lee V Ravichandran and S Supramaniam ldquoBounds forthe second Hankel determinant of certain univalent functionsrdquoJournal of Inequalities and Applications vol 2013 article 2812013
[6] D Vamshee Krishna andT Ramreddy ldquoHankel determinant forstarlike and convex functions of order alphardquo Tbilisi Mathemat-ical Journal vol 5 no 1 pp 65ndash76 2012
[7] C Selvaraj and T R K Kumar ldquoSecond Hankel determinantfor certain classes of analytic functionsrdquo International Journalof Applied Mathematics vol 28 no 1 pp 37ndash50 2015
[8] W Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis Conference Proceedings and LectureNotes in Analysis Vol 1 pp 157ndash169 International PressTianjin China 1992
[9] J Sokoł and J Stankiewicz ldquoRadius of convexity of some sub-classes of strongly starlike functionsrdquoZeszytyNaukowe Politech-niki Rzeszowskiej Matematyka vol 19 pp 101ndash105 1996
[10] M-S Liu J-F Xu and M Yang ldquoUpper bound of second Han-kel determinant for certain subclasses of analytic functionsrdquoAbstract and Applied Analysis vol 2014 Article ID 603180 10pages 2014
[11] A K Mishra and P Gochhayat ldquoSecond Hankel determinantfor a class of analytic functions defined by fractional deriva-tiverdquo International Journal of Mathematics and MathematicalSciences vol 2008 Article ID 153280 10 pages 2008
[12] J W Noonan and D K Thomas ldquoOn the Hankel determinantsof areally mean 119901-valent functionsrdquo Proceedings of the LondonMathematical Society vol 25 pp 503ndash524 1972
[13] G Singh ldquoHankel determinant for a new subclass of analyticfunctionsrdquo Scientia Magna vol 8 no 4 pp 61ndash65 2012
[14] G Singh ldquoHankel determinant for analytic functions withrespect to other pointsrdquo Engineering Mathematics Letters vol2 no 2 pp 115ndash123 2013
[15] T Hayami and S Owa ldquoGeneralized hankel determinant forcertain classesrdquo International Journal of Mathematical Analysisvol 4 no 52 pp 2573ndash2585 2010
[16] K O Babalola ldquoOnH3(1) Hankel determinants for some classes
of univalent functionsrdquo in Inequality Theory and Applicationsvol 6 17 Nova Science Publishers New York NY USA 2010
[17] G Shanmugam B Adolf Stephen and K O Babalola ldquoThirdHankel determinant for 120572-starlike functionsrdquo Gulf Journal ofMathematics vol 2 no 2 pp 107ndash113 2014
[18] P T Mocanu ldquoUne proprit de convexit gnralise dans la thoriede la representation conformerdquoMathematica (Cluj) vol 11 no34 pp 127ndash133 1969
[19] M S Robertson ldquoOn the coefficients of a typically-real func-tionrdquo Bulletin of the American Mathematical Society vol 41 no8 pp 565ndash572 1935
[20] D J Hallenbeck and T H MacGregor Linear Problems andConvexity Techniques in Geometric Function Theory Mono-graphs and Studies in Mathematics 22 Pitman AdavancedPublishing Program Boston Mass USA 1984
[21] W Ma ldquoGeneralized Zalcman conjecture for starlike and typ-ically real functionsrdquo Journal of Mathematical Analysis andApplications vol 234 no 1 pp 328ndash339 1999
[22] W Szapiel Extremal problems for convex sets Applications toholomorphic functions [Dissertation XXXVII] UMCS PressLublin Poland reviewer J Krzyz 1986 (Polish)
[23] W Szapiel ldquoExtreme points of convex sets (II) Inuence of nor-malization on integral representationrdquo Bulletin de lrsquoAcademiePolonaise des Sciences Serie des SciencesMathematiques vol 29pp 535ndash544 1981
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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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
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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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
2 Class 119879 and the Hankel Determinants fora Selected Functions in 119879
A function 119891 isin A that satisfies the condition Im 119911 Im119891(119911) ge
0 for 119911 isin Δ is called a typically real function Let 119879 denotethe class of all typically real functions Robertson [19] provedthat 119891 isin 119879 if and only if there exists a probability measure 120583on [minus1 1] such that the following formula holds
119891 (119911) = int
1
minus1
119911
1 minus 2119911119905 + 1199112119889120583 (119905) (2)
The coefficients of a function 119891(119911) = 119911 + suminfin
119899=2119886119899119911119899isin 119879 can
be written as follows
119886119899= int
1
minus1
sin (119899 arccos 119905)sin (arccos 119905)
119889120583 (119905) = int
1
minus1
119880119899minus1
(119905) 119889120583 (119905)
119899 = 1 2
(3)
The functions 119880119899(119905) 119899 = 1 2 which appear in the above
formula are the well-known Chebyshev polynomials of thesecond kind
Since all coefficients of 119891 isin 119879 are real we look for thelower and the upper bounds of119867
2(2) instead of the bound of
|1198672(2)| At the beginning let us look at a few examples
Example 1 All the functions 119891119905(119911) = 119911(1 minus 2119911119905 + 119911
2) 119905 isin
[minus1 1] are in 119878lowast Since 119891119905(119911) = 119911 + 2119905119911
2+ (41199052minus 1)1199113+ (81199053minus
4119905)1199114+sdot sdot sdot we have119867
2(2) = minus1 for each 119905 isin [minus1 1]Moreover
1198672(119899) = 119880
119899minus1(119905)119880119899+1
(119905) minus 119880119899(119905)2 This and the Turan identity
for Chebyshev polynomials 119880119899(119905) result in 119867
2(119899) = minus1 for
each 119899 = 2 3
Example 2 For a function 119891(119911) = 119911(1 + 1199112)(1 minus 119911
2)2 having
the Taylor series expansion 119891(119911) = 119911 + 31199113+ 51199115+ sdot sdot sdot there
is1198672(119899) = minus(119899 + 1)
2 for even 119899 and1198672(119899) = (119899 + 1)
2minus 1 for
odd 119899 In this case the function 119891 is not univalent the boundof |1198672(2)| is much greater than 1 the value of the second
Hankel determinant for star-like functions or close-to-convexfunctions
Example 3 EveryHankel determinant1198672(119899) 119899 = 1 2 for
a function 119891(119911) = log(1(1 minus 119911)) = 119911+ (12)1199112+ (13)119911
3+ sdot sdot sdot
is positive Namely1198672(119899) = 1119899(119899 + 1)
2(119899 + 2)
For a given class 119860 sub A we denote by Ω119899(119860) 119899 ge 1
the region of variability of three succeeding coefficients offunctions in 119860 that is the set (119886
119899(119891) 119886119899+1
(119891) 119886119899+2
(119891))
119891 isin 119860 As it is seen in (3) the coefficients of typicallyreal functions are the Stieltjes integrals of the Chebyshevpolynomials of the second kind with respect to a probabilitymeasure HenceΩ
119899(119879) is the closed convex hull of the curve
120574 [minus1 1] ni 119905 rarr (119880119899minus1
(119905) 119880119899(119905) 119880119899+1
(119905)) (see eg [20])
Lemma 4 The functional 119879 ni 119891 rarr 1198672(119899) 119899 ge 2 attains its
extreme values on the boundary of Ω119899(119879)
Proof The only critical point of ℎ(119909 119910 119911) = 119909119911 minus 1199102 where
119909 = 119886119899 119910 = 119886
119899+1 and 119911 = 119886
119899+2 is (0 0 0) But ℎ(0 0 0) = 0
Since ℎ may be positive as well as negative for (119909 119910 119911) isin
Ω119899(119879) (see Examples 1 and 3) it means that the extreme
values of ℎ are attained on the boundary ofΩ119899(119879)
3 Bounds of 1198672(2) in 119879
In [21] Ma proved so-called generalized Zalcman conjecturefor the class 119879
1003816100381610038161003816119886119899119886119898 minus 119886119899+119898minus1
1003816100381610038161003816
le
119899 + 1 119898 = 2 119899 = 2 4 6
119898 + 1 119899 = 2 119898 = 2 4 6
(119899 minus 1) (119898 minus 1) otherwise
(4)
We apply this result to prove the following
Theorem 5 If 119891 isin 119879 then |1198672(2)| le 9
Proof The result of Ma and the triangle inequality result in1003816100381610038161003816100381611988621198864minus 1198863
210038161003816100381610038161003816le100381610038161003816100381611988621198864 minus 119886
5
1003816100381610038161003816 +100381610038161003816100381610038161198865minus 1198863
210038161003816100381610038161003816le 5 + 4 = 9 (5)
This result is sharp the equality holds for 119891(119911) = 119911(1 +
1199112)(1 minus 119911
2)2 Furthermore we can see the following
Corollary 6 For 119879 one has
min 1198672(2) 119891 isin 119879 = minus9 (6)
For our next theorem let us cite two results First one isthe obvious conclusion from the Caratheodory theorem andthe Krein-Milman theorem We assume that 119883 is a compactHausdorff space and
119869120583= int119883
119869 (119905) 119889120583 (119905) (7)
Theorem A (see [22 Thm 140]) If 119869 119883 rarr R119899 is continu-ous then the convex hull of 119869(119883) is a compact set and it coincideswith the set 119869
120583 120583 isin 119875
119883 |supp(120583)| le 119899
In the above the symbols 119875119883and |supp(120583)| stand for the
set of probability measures on 119883 and the cardinality of thesupport of 120583 respectively
It means that 120583 is atomic measure having at most 119899 stepsMore precise information about the relation between themeasure and the convex hull is presented in the followingtheorem In what follows ⟨ 119886 ⟩ means the scalar product of119886 and
Theorem B (see [22 Thm 149]) Let 119869 [120572 120573] rarr R119899 becontinuous Suppose that there exists a positive integer 119896 suchthat for each nonzero 119901 in R119899 the number of solutions of anyequation ⟨
997888997888rarr119869(119905) ⟩ = const 120572 le 119905 le 120573 is not greater than 119896
Abstract and Applied Analysis 3
Then for every 120583 isin 119875[120572120573]
such that 119869120583belongs to the boundary
of the convex hull of 119869([120572 120573]) the following statements aretrue
(1) If 119896 = 2119898 then
(a) |supp(120583)| le 119898 or(b) |supp(120583)| = 119898 + 1 and 120572 120573 sub supp(120583)
(2) If 119896 = 2119898 + 1 then
(a) |supp(120583)| le 119898 or(b) |supp(120583)| = 119898 + 1 and one of the points 120572 and 120573
belongs to supp(120583)
This theorem in slightly modified version was publishedin [23] as Lemma 2
Putting 119869(119905) = [1198801(119905) 1198802(119905) 1198803(119905)] 119905 isin [minus1 1] and =
[1199011 1199012 1199013] we can see that any equation of the form
11990111198801(119905) + 119901
21198802(119905) + 119901
31198803(119905) = const 119905 isin [minus1 1] (8)
is equivalent to 1198823(119905) = const where 119882
3(119905) is a polynomial
of degree 3 Hence (8) has at most 3 solutions According toTheorem B the boundary of the convex hull of 119869([minus1 1]) isdetermined by atomic measures 120583 for which support consistsof at most 2 points Moreover one of them has to be minus1 or 1We have proved the following
Lemma7 Theboundary ofΩ2(119879) consists of points (119886
2 1198863 1198864)
that correspond to the following functions
119891 (119911) = 120572119911
1 minus 2119911119905 + 1199112+ (1 minus 120572)
119911
(1 minus 119911)2
120572 isin [0 1] 119905 isin [minus1 1]
(9)
or
119891 (119911) = 120572119911
1 minus 2119911119905 + 1199112+ (1 minus 120572)
119911
(1 + 119911)2
120572 isin [0 1] 119905 isin [minus1 1]
(10)
Now we are ready to prove the following
Theorem 8 For 119879 one has
max 1198672(2) 119891 isin 119879 = 1 (11)
Proof By Lemma 7 it is enough to take functions given by(9) or (10) Consider the following
(I) Function (9) has the series expansion
119891 (119911) = 119911 + 2 (1 minus 120572 + 120572119905) 1199112
+ [3 (1 minus 120572) + (41199052minus 1) 120572] 119911
3
+ 4 [1 minus 120572 + (21199053minus 119905) 120572] 119911
4+ sdot sdot sdot
(12)
Hence1198672(2) = 119892
1(120572 119905) where
1198921(120572 119905) = 8 (1 minus 120572 + 120572119905) [1 minus 120572 + (2119905
3minus 119905) 120572]
minus (41199052120572 minus 4120572 + 3)
2
120572 isin [0 1] 119905 isin [minus1 1]
(13)
From
1205971198921
120597120572= 8 (1 minus 119905
2) (2 (2 minus 120572) 119905
2+ 1 minus 2120572)
1205971198921
120597119905= 16120572 (120572 minus 1) (4119905
2minus 3119905 + 3)
(14)
it follows that the critical points of 1198921are as follows (0 minus1)
(0 1) (1 minus1) (1 1) (1 minus1radic2) (1 1radic2) and (12 0)Among these points only (12 0) lies inside the set [0 1] times[minus1 1]
If 120572 = 0 or 120572 = 1 then functions (9) coincide with 119891119905
from Example 1 If 119905 = 1 then 119891(119911) = 119911(1 minus 119911)2 In each
case 1198672(2) = minus1 For 119905 = minus1 function (9) takes the form
119891(119911) = 120572(119911(1 + 119911)2) + (1 minus 120572)(119911(1 minus 119911)
2) Then 119867
2(2) =
8(1 minus 2120572)2minus 9 le minus1
If 120572 = 12 and 119905 = 0 we have1198672(2) = 1 It means that the
greatest value of1198672(2) for functions given by (9) is equal to 1
The extremal function is
119891 (119911) =1
2[
119911
1 + 1199112+
119911
(1 minus 119911)2]
= 119911 + 1199112+ 1199113+ 21199114+ 31199115+ sdot sdot sdot
(15)
(II) For functions (10)1198672(2) is equal to 119892
2(120572 119905) where
1198922(120572 119905) = 8 (minus1 + 120572 + 120572119905) [minus1 + 120572 + (2119905
3minus 119905) 120572]
minus (41199052120572 minus 4120572 + 3)
2
120572 isin [0 1] 119905 isin [minus1 1]
(16)
Moreover 1198922(120572 119905) = 119892
1(120572 minus119905) Taking into account the sym-
metry of the range of variability of 119905 we obtain the sameresult as above also for functions defined by (10)The extremalfunction is
119891 (119911) =1
2[
119911
1 + 1199112+
119911
(1 + 119911)2]
= 119911 minus 1199112+ 1199113minus 21199114+ 31199115+ sdot sdot sdot
(17)
4 Bounds of 1198672(3) in 119879
The proof of the following theorem is obvious
Theorem 9 If 119899 is odd then
max 1198672(119899) 119891 isin 119879 = 119899 (119899 + 2) (18)
Hence one has the following
4 Abstract and Applied Analysis
Corollary 10 For 119879 one has
max 1198672(3) 119891 isin 119879 = 15 (19)
In similar way as it was done for Lemma 7 one can provethe following
Lemma 11 The boundary of Ω3(119879) consists of points (119886
3
1198864 1198865) that correspond to the following functions
119891 (119911) = 120572119911
1 minus 21199111199051+ 1199112
+ (1 minus 120572)119911
1 minus 21199111199052+ 1199112
120572 isin [0 1] 1199051 1199052 isin [minus1 1]
(20)
or
119891 (119911) = 120572119911
(1 + 119911)2+ 120573
119911
(1 minus 119911)2
+ (1 minus 120572 minus 120573)119911
1 minus 2119911119905 + 1199112
120572 120573 isin [0 1] 120572 + 120573 le 1 119905 isin [minus1 1]
(21)
Theorem 12 For 119879 one has
min 1198672(3) 119891 isin 119879 = minus
4
3+8
9
radic6 = minus351 (22)
Proof By Lemma 11 it suffices to discuss functions given by(20) or (21) Consider the following
(I) For functions (20) we have
1198863= (41199051
2minus 1) 120572 + (4119905
2
2minus 1) (1 minus 120572)
1198864= (81199051
3minus 41199051) 120572 + (8119905
2
3minus 41199052) (1 minus 120572)
1198865= (16119905
1
4minus 121199051
2+ 1) 120572
+ (161199052
4minus 121199052
2+ 1) (1 minus 120572)
(23)
and hence applying the Turan identity1198672(3) = 119892
3(120572 1199051 1199052)
where
1198923(120572 1199051 1199052)
= minus1205722minus (1 minus 120572)
2
+ 2120572 (1 minus 120572) [8 (1199051minus 1199052)2
(1 minus 1199051
2) (1 minus 119905
2
2) minus 1]
120572 isin [0 1] 1199051 1199052 isin [minus1 1]
(24)
The expression in brackets is greater than or equal to minus1for all 119905
1 1199052isin [minus1 1] Hence
1198672(3) ge minus120572
2minus (1 minus 120572)
2minus 2120572 (1 minus 120572) = minus1 (25)
(II) If function 119891 is given by (21) then
1198863= 3 (120572 + 120573) + (4119905
2minus 1) (1 minus 120572 minus 120573)
1198864= 4 (120573 minus 120572) + (8119905
3minus 4119905) (1 minus 120572 minus 120573)
1198865= 5 (120572 + 120573) + (16119905
4minus 121199052+ 1) (1 minus 120572 minus 120573)
(26)
Using the Turan identity it follows that 1198672(3) = 119892
4(120572 120573 119905)
where
1198924(120572 120573 119905) = minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)
2
+ 64120572120573
+ 2 (1 minus 120572 minus 120573) 119902 (119905)
119902 (119905) = (120572 + 120573) (241199054minus 81199052minus 1)
+ (120572 minus 120573) (321199053minus 16119905)
(27)
under the assumptions 120572 120573 isin [0 1] 120572+120573 le 1 and 119905 isin [minus1 1]Let 120572 and 120573 be fixed Since
120597119902
120597119905= 8 (6119905
2minus 1) [119905 (120572 + 120573) + 120572 minus 120573] (28)
the critical points of 119902 are as follows minus1radic6 1radic6 and (120573 minus
120572)(120572 + 120573) It is easily seen that all these points are in [minus1 1]Therefore
min 119902 (119905) 119905 isin [minus1 1] = min119902 (minus1) 119902 ( minus1
radic6
)
119902 (1
radic6
) 119902 (1) 119902 (120573 minus 120572
120572 + 120573)
(29)
For 119905 = minus1 or 119905 = 1 the functions given by (21) have theform
119891 (119911) = 119911
(1 + 119911)2+ 120573
119911
(1 minus 119911)2 + 120573 = 1 (30)
One can show directly from formula (30) that
1198672(3) = minus1 + 64 (1 minus ) ge minus1 (31)
For 119905 = (120573 minus 120572)(120572 + 120573) there is
1198672(3) = minus1 +
64120572120573 [4 (120572 + 120573) 120572120573 + (120573 minus 120572)2
]
(120572 + 120573)3
(32)
hence
1198672(3) ge minus1 (33)
If 119905 = minus1radic6 or 119905 = 1radic6 then1198672(3) is equal to
1198672(3)
= minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)2
+ 64120572120573
+ (1 minus 120572 minus 120573) [minus10
3(120572 + 120573) minus
64
3radic6
(120572 minus 120573)]
(34)
or
1198672(3)
= minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)2
+ 64120572120573
+ (1 minus 120572 minus 120573) [minus10
3(120572 + 120573) +
64
3radic6
(120572 minus 120573)]
(35)
Abstract and Applied Analysis 5
respectively Without loss of generality we can assume that120572 ge 120573 Then while looking for the minimum value of119867
2(3)
we can restrict the research to the first stated above case (sinceexpression (35) is not less than expression (34))
Transforming (35) we obtain
1198672(3) = minus1 + 64120572120573 minus
4
3(120572 + 120573) (1 minus 120572 minus 120573)
+64
3radic6
(120573 minus 120572) (1 minus 120572 minus 120573)
(36)
Taking the smallest possible 120573 (ie 120573 = 0) the second andthe forth component of this expression will not increase Thevalue of the third component does not depend only on 120573 infact it depends on 120572 + 120573 For this reason we can take 120573 = 0Combining these facts it yields that
1198672(3) ge minus1 minus
4
3120572 (1 minus 120572) minus
64
3radic6
120572 (1 minus 120572) (37)
The smallest value of the right hand side of this inequality isachieved for 120572 = 12 In this case
1198672(3) ge minus1 minus
1
3minus
16
3radic6
= minus4
3minus8
9
radic6 = minus351 (38)
Combining two parts of the proof we obtain the conclu-sion of the theorem Furthermore the above shows that theextremal functions are
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 21199111199050+ 1199112
]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 + 21199111199050+ 1199112
]
(39)
where 1199050= 1radic6
5 Bounds of 1198672(119899) 119899 ge 4 in 119879
It is easily seen that 1198672(119899) le 119899(119899 + 2) for any typically real
function ByTheorem 9 this estimate is sharp providing that119899 is an odd integer At the beginning of this section we willprove the following
Theorem 13 For 119879 one has
min 119886119899119886119899+2
119891 isin 119879 = minus1 (40)
Proof Thecoefficients of the series expansion of function119891 isin
119879 can be written as follows
119886119899= int
120587
0
sin (119899120579)sin 120579
119889] (120579) ] isin 119875[0120587]
(41)
Hence
119886119899119886119899+2
= int
120587
0
sin ((119899 + 1) 120579 minus 120579)
sin 120579119889] (120579)
sdot int
120587
0
sin ((119899 + 1) 120579 + 120579)
sin 120579119889] (120579)
= (int
120587
0
sin (119899 + 1) 120579
sin 120579cos 120579 119889] (120579))
2
minus (int
120587
0
cos (119899 + 1) 120579 119889] (120579))2
(42)
Since
int
120587
0
cos (119899 + 1) 120579 119889] (120579) le int
120587
0
119889] (120579) = 1 (43)
we obtain
119886119899119886119899+2
ge minus1 (44)
In order to prove that the estimate is sharp let us take themeasure ] for which support satisfies condition (119899 + 1)120579 =
120587 This measure corresponds to the function 119891(119911) = 119911(1 minus
2119911cos(120587(119899 + 1)) + 1199112)
Observe that 119886119899119886119899+2
= minus1 holds not only for the measurestated above Namely the value minus1 in (42) is taken also if (119899 +1)120579 = 119896120587 where 119896 is any positive integer less than or equal to119899 From this we conclude that the support of the measure has119899 points 120579
119896= 119896120587(119899 + 1) with weights 120572
119896 119896 = 1 2 119899 such
that sum119899119896=1
120572119896= 1
The weights satisfy
(
119899
sum
119896=1
120572119896(minus1)119896)
2
= 1 (45)
Indeed if the support of ] consists of 119899 points then 119891 takesthe form
119891 (119911) =
119899
sum
119896=1
120572119896
119911
1 minus 2119911 cos 120579119896+ 1199112
(46)
Using trigonometric identities we obtain
119886119899=
119899
sum
119896=1
120572119896(minus1)119896+1
119886119899+2
=
119899
sum
119896=1
120572119896(minus1)119896
(47)
which results in (45)Connecting (45) andsum119899
119896=1120572119896= 1we conclude that119891 is of
the form
119891 (119911) =
119899
sum
119896=1119896 is odd120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112(48)
6 Abstract and Applied Analysis
Table 1 The bounds of the Hankel determinants for functions defined by (51)
Bounds of1198672(119899) Equality in the lower bound for Equality in the upper bound for
minus9 le 1198672(2) le 1 119905 = 1 119905 = 0
minus351 le 1198672(3) le 15 119905 = 0 40 119905 = 1
minus25 le 1198672(4) le 446 119905 = 1 119905 = 061
minus784 le 1198672(5) le 35 119905 = 0 72 119905 = 1
minus49 le 1198672(6) le 967 119905 = 1 119905 = 079
or
119891 (119911) =
119899
sum
119896=1119896 is even120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112 (49)
Itmeans that for even 119899 the support of ] consists of 1198992 pointsand for even 119899 the number of points of the support of ] is equalto (119899 + 1)2 or (119899 minus 1)2
Taking into account |119886119899+1
| le 119899 + 1 and Theorem 13 weobtain the following
Theorem 14 For 119879 one has
1198672(119899) ge minus (119899 + 1)
2minus 1 (50)
Unfortunately this bound is not sharp However thefollowing can be conjectured
Conjecture 15 For any positive integer 119899 the following esti-mate 119867
2(119899) ge minus(119899 + 1)
2 holds Moreover this bound is sharpfor even 119899
This conjecture is supported by the facts that in thetheorems concerning119867
2(2) and119867
2(3) the extremal functions
are of the form
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 2119911119905 + 1199112]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 minus 2119911119905 + 1199112]
(51)
for appropriately taken 119905 isin [minus1 1] The exact bounds ofthe Hankel determinants for these functions are collected inTable 1 They were obtained numerically
6 Remarks Concerning 1198672(119899) in 119879
(2)
In class 119879 we discuss subclass 119879(2) consisting of the functionswhich are odd The definition of this class is
119879(2)
= 119891 isin 119879 119891 (minus119911) = minus119891 (119911) 119911 isin Δ (52)
For 119891 isin 119879(2) the representation formula similar to (2) is
valid Namely
119891 (119911) = int
1
minus1
119911 (1 + 1199112)
(1 + 1199112)2
minus 411991121199052119889] (119905) ] isin 119875
[minus11] (53)
Function 119891 has the Taylor series expansion
119891 (119911) = sum
119899 is odd119886119899119911119899 119886119899= int
1
minus1
119880119899minus1
(119905) 119889] (119905) (54)
The following inequalities are obvious
minus (119899 + 1)2le 1198672(119899) le 0 for even 119899
1198672(119899) le (119899 + 1)
2minus 1 for odd 119899
(55)
equalities hold for 119891(119911) = 119911(1 + 1199112)(1 minus 119911
2)2
For a given class 119860 sub A let us denote by Ψ119899(119860) 119899 ge 1
the set (119886119899 119886119899+2
) 119891 isin 119860 From (53) it follows that Ψ119899(119879) is
the closed convex hull of the curve
120582 [minus1 1] ni 119905 997888rarr (119880119899minus1
(119905) 119880119899+1
(119905)) (56)
FromTheorem 13 and from the equivalence
(119886119899 119886119899+2
) isin Ψ119899(119879) lArrrArr (119886
119899 119886119899+2
) isin Ψ119899(119879(2)) (57)
we get
min 119886119899119886119899+2
119891 isin 119879(2) = minus1 (58)
Hence for odd 119899 we know that
min 1198672(119899) 119891 isin 119879
(2) ge minus1 (59)
The equality holds for functions (48) or (49) providing that120572119896
= 120572119899+1minus119896
Then connecting the components of theseformulae in pairs we obtain
120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112
+ 120572119899+1minus119896
119911
1 minus 2119911 cos ((119899 + 1 minus 119896) 120587 (119899 + 1)) + 1199112
= 2120572119896
119911 (1 + 1199112)
(1 + 1199112)2
minus 41199112cos2 (119896120587 (119899 + 1))
(60)
With help of the argument given in the proof of Theorem 13we eventually obtain the odd functions for which 119886
119899119886119899+2
=
minus1
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 7
References
[1] C Pommerenke ldquoOn the coefficients and Hankel determinantsof univalent functionsrdquo Journal of the London MathematicalSociety vol 41 pp 111ndash122 1966
[2] C Pommerenke ldquoOn the Hankel determinants of univalentfunctionsrdquoMathematika vol 14 pp 108ndash112 1967
[3] A Janteng S A Halim and M Darus ldquoCoefficient inequalityfor a function whose derivative has a positive real partrdquo Journalof Inequalities in Pure and Applied Mathematics vol 7 no 2article 50 2006
[4] A Janteng S A Halim andMDarus ldquoHankel determinant forstarlike and convex functionsrdquo International Journal of Mathe-matical Analysis vol 1 no 13 pp 619ndash625 2007
[5] S K Lee V Ravichandran and S Supramaniam ldquoBounds forthe second Hankel determinant of certain univalent functionsrdquoJournal of Inequalities and Applications vol 2013 article 2812013
[6] D Vamshee Krishna andT Ramreddy ldquoHankel determinant forstarlike and convex functions of order alphardquo Tbilisi Mathemat-ical Journal vol 5 no 1 pp 65ndash76 2012
[7] C Selvaraj and T R K Kumar ldquoSecond Hankel determinantfor certain classes of analytic functionsrdquo International Journalof Applied Mathematics vol 28 no 1 pp 37ndash50 2015
[8] W Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis Conference Proceedings and LectureNotes in Analysis Vol 1 pp 157ndash169 International PressTianjin China 1992
[9] J Sokoł and J Stankiewicz ldquoRadius of convexity of some sub-classes of strongly starlike functionsrdquoZeszytyNaukowe Politech-niki Rzeszowskiej Matematyka vol 19 pp 101ndash105 1996
[10] M-S Liu J-F Xu and M Yang ldquoUpper bound of second Han-kel determinant for certain subclasses of analytic functionsrdquoAbstract and Applied Analysis vol 2014 Article ID 603180 10pages 2014
[11] A K Mishra and P Gochhayat ldquoSecond Hankel determinantfor a class of analytic functions defined by fractional deriva-tiverdquo International Journal of Mathematics and MathematicalSciences vol 2008 Article ID 153280 10 pages 2008
[12] J W Noonan and D K Thomas ldquoOn the Hankel determinantsof areally mean 119901-valent functionsrdquo Proceedings of the LondonMathematical Society vol 25 pp 503ndash524 1972
[13] G Singh ldquoHankel determinant for a new subclass of analyticfunctionsrdquo Scientia Magna vol 8 no 4 pp 61ndash65 2012
[14] G Singh ldquoHankel determinant for analytic functions withrespect to other pointsrdquo Engineering Mathematics Letters vol2 no 2 pp 115ndash123 2013
[15] T Hayami and S Owa ldquoGeneralized hankel determinant forcertain classesrdquo International Journal of Mathematical Analysisvol 4 no 52 pp 2573ndash2585 2010
[16] K O Babalola ldquoOnH3(1) Hankel determinants for some classes
of univalent functionsrdquo in Inequality Theory and Applicationsvol 6 17 Nova Science Publishers New York NY USA 2010
[17] G Shanmugam B Adolf Stephen and K O Babalola ldquoThirdHankel determinant for 120572-starlike functionsrdquo Gulf Journal ofMathematics vol 2 no 2 pp 107ndash113 2014
[18] P T Mocanu ldquoUne proprit de convexit gnralise dans la thoriede la representation conformerdquoMathematica (Cluj) vol 11 no34 pp 127ndash133 1969
[19] M S Robertson ldquoOn the coefficients of a typically-real func-tionrdquo Bulletin of the American Mathematical Society vol 41 no8 pp 565ndash572 1935
[20] D J Hallenbeck and T H MacGregor Linear Problems andConvexity Techniques in Geometric Function Theory Mono-graphs and Studies in Mathematics 22 Pitman AdavancedPublishing Program Boston Mass USA 1984
[21] W Ma ldquoGeneralized Zalcman conjecture for starlike and typ-ically real functionsrdquo Journal of Mathematical Analysis andApplications vol 234 no 1 pp 328ndash339 1999
[22] W Szapiel Extremal problems for convex sets Applications toholomorphic functions [Dissertation XXXVII] UMCS PressLublin Poland reviewer J Krzyz 1986 (Polish)
[23] W Szapiel ldquoExtreme points of convex sets (II) Inuence of nor-malization on integral representationrdquo Bulletin de lrsquoAcademiePolonaise des Sciences Serie des SciencesMathematiques vol 29pp 535ndash544 1981
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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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
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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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
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Then for every 120583 isin 119875[120572120573]
such that 119869120583belongs to the boundary
of the convex hull of 119869([120572 120573]) the following statements aretrue
(1) If 119896 = 2119898 then
(a) |supp(120583)| le 119898 or(b) |supp(120583)| = 119898 + 1 and 120572 120573 sub supp(120583)
(2) If 119896 = 2119898 + 1 then
(a) |supp(120583)| le 119898 or(b) |supp(120583)| = 119898 + 1 and one of the points 120572 and 120573
belongs to supp(120583)
This theorem in slightly modified version was publishedin [23] as Lemma 2
Putting 119869(119905) = [1198801(119905) 1198802(119905) 1198803(119905)] 119905 isin [minus1 1] and =
[1199011 1199012 1199013] we can see that any equation of the form
11990111198801(119905) + 119901
21198802(119905) + 119901
31198803(119905) = const 119905 isin [minus1 1] (8)
is equivalent to 1198823(119905) = const where 119882
3(119905) is a polynomial
of degree 3 Hence (8) has at most 3 solutions According toTheorem B the boundary of the convex hull of 119869([minus1 1]) isdetermined by atomic measures 120583 for which support consistsof at most 2 points Moreover one of them has to be minus1 or 1We have proved the following
Lemma7 Theboundary ofΩ2(119879) consists of points (119886
2 1198863 1198864)
that correspond to the following functions
119891 (119911) = 120572119911
1 minus 2119911119905 + 1199112+ (1 minus 120572)
119911
(1 minus 119911)2
120572 isin [0 1] 119905 isin [minus1 1]
(9)
or
119891 (119911) = 120572119911
1 minus 2119911119905 + 1199112+ (1 minus 120572)
119911
(1 + 119911)2
120572 isin [0 1] 119905 isin [minus1 1]
(10)
Now we are ready to prove the following
Theorem 8 For 119879 one has
max 1198672(2) 119891 isin 119879 = 1 (11)
Proof By Lemma 7 it is enough to take functions given by(9) or (10) Consider the following
(I) Function (9) has the series expansion
119891 (119911) = 119911 + 2 (1 minus 120572 + 120572119905) 1199112
+ [3 (1 minus 120572) + (41199052minus 1) 120572] 119911
3
+ 4 [1 minus 120572 + (21199053minus 119905) 120572] 119911
4+ sdot sdot sdot
(12)
Hence1198672(2) = 119892
1(120572 119905) where
1198921(120572 119905) = 8 (1 minus 120572 + 120572119905) [1 minus 120572 + (2119905
3minus 119905) 120572]
minus (41199052120572 minus 4120572 + 3)
2
120572 isin [0 1] 119905 isin [minus1 1]
(13)
From
1205971198921
120597120572= 8 (1 minus 119905
2) (2 (2 minus 120572) 119905
2+ 1 minus 2120572)
1205971198921
120597119905= 16120572 (120572 minus 1) (4119905
2minus 3119905 + 3)
(14)
it follows that the critical points of 1198921are as follows (0 minus1)
(0 1) (1 minus1) (1 1) (1 minus1radic2) (1 1radic2) and (12 0)Among these points only (12 0) lies inside the set [0 1] times[minus1 1]
If 120572 = 0 or 120572 = 1 then functions (9) coincide with 119891119905
from Example 1 If 119905 = 1 then 119891(119911) = 119911(1 minus 119911)2 In each
case 1198672(2) = minus1 For 119905 = minus1 function (9) takes the form
119891(119911) = 120572(119911(1 + 119911)2) + (1 minus 120572)(119911(1 minus 119911)
2) Then 119867
2(2) =
8(1 minus 2120572)2minus 9 le minus1
If 120572 = 12 and 119905 = 0 we have1198672(2) = 1 It means that the
greatest value of1198672(2) for functions given by (9) is equal to 1
The extremal function is
119891 (119911) =1
2[
119911
1 + 1199112+
119911
(1 minus 119911)2]
= 119911 + 1199112+ 1199113+ 21199114+ 31199115+ sdot sdot sdot
(15)
(II) For functions (10)1198672(2) is equal to 119892
2(120572 119905) where
1198922(120572 119905) = 8 (minus1 + 120572 + 120572119905) [minus1 + 120572 + (2119905
3minus 119905) 120572]
minus (41199052120572 minus 4120572 + 3)
2
120572 isin [0 1] 119905 isin [minus1 1]
(16)
Moreover 1198922(120572 119905) = 119892
1(120572 minus119905) Taking into account the sym-
metry of the range of variability of 119905 we obtain the sameresult as above also for functions defined by (10)The extremalfunction is
119891 (119911) =1
2[
119911
1 + 1199112+
119911
(1 + 119911)2]
= 119911 minus 1199112+ 1199113minus 21199114+ 31199115+ sdot sdot sdot
(17)
4 Bounds of 1198672(3) in 119879
The proof of the following theorem is obvious
Theorem 9 If 119899 is odd then
max 1198672(119899) 119891 isin 119879 = 119899 (119899 + 2) (18)
Hence one has the following
4 Abstract and Applied Analysis
Corollary 10 For 119879 one has
max 1198672(3) 119891 isin 119879 = 15 (19)
In similar way as it was done for Lemma 7 one can provethe following
Lemma 11 The boundary of Ω3(119879) consists of points (119886
3
1198864 1198865) that correspond to the following functions
119891 (119911) = 120572119911
1 minus 21199111199051+ 1199112
+ (1 minus 120572)119911
1 minus 21199111199052+ 1199112
120572 isin [0 1] 1199051 1199052 isin [minus1 1]
(20)
or
119891 (119911) = 120572119911
(1 + 119911)2+ 120573
119911
(1 minus 119911)2
+ (1 minus 120572 minus 120573)119911
1 minus 2119911119905 + 1199112
120572 120573 isin [0 1] 120572 + 120573 le 1 119905 isin [minus1 1]
(21)
Theorem 12 For 119879 one has
min 1198672(3) 119891 isin 119879 = minus
4
3+8
9
radic6 = minus351 (22)
Proof By Lemma 11 it suffices to discuss functions given by(20) or (21) Consider the following
(I) For functions (20) we have
1198863= (41199051
2minus 1) 120572 + (4119905
2
2minus 1) (1 minus 120572)
1198864= (81199051
3minus 41199051) 120572 + (8119905
2
3minus 41199052) (1 minus 120572)
1198865= (16119905
1
4minus 121199051
2+ 1) 120572
+ (161199052
4minus 121199052
2+ 1) (1 minus 120572)
(23)
and hence applying the Turan identity1198672(3) = 119892
3(120572 1199051 1199052)
where
1198923(120572 1199051 1199052)
= minus1205722minus (1 minus 120572)
2
+ 2120572 (1 minus 120572) [8 (1199051minus 1199052)2
(1 minus 1199051
2) (1 minus 119905
2
2) minus 1]
120572 isin [0 1] 1199051 1199052 isin [minus1 1]
(24)
The expression in brackets is greater than or equal to minus1for all 119905
1 1199052isin [minus1 1] Hence
1198672(3) ge minus120572
2minus (1 minus 120572)
2minus 2120572 (1 minus 120572) = minus1 (25)
(II) If function 119891 is given by (21) then
1198863= 3 (120572 + 120573) + (4119905
2minus 1) (1 minus 120572 minus 120573)
1198864= 4 (120573 minus 120572) + (8119905
3minus 4119905) (1 minus 120572 minus 120573)
1198865= 5 (120572 + 120573) + (16119905
4minus 121199052+ 1) (1 minus 120572 minus 120573)
(26)
Using the Turan identity it follows that 1198672(3) = 119892
4(120572 120573 119905)
where
1198924(120572 120573 119905) = minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)
2
+ 64120572120573
+ 2 (1 minus 120572 minus 120573) 119902 (119905)
119902 (119905) = (120572 + 120573) (241199054minus 81199052minus 1)
+ (120572 minus 120573) (321199053minus 16119905)
(27)
under the assumptions 120572 120573 isin [0 1] 120572+120573 le 1 and 119905 isin [minus1 1]Let 120572 and 120573 be fixed Since
120597119902
120597119905= 8 (6119905
2minus 1) [119905 (120572 + 120573) + 120572 minus 120573] (28)
the critical points of 119902 are as follows minus1radic6 1radic6 and (120573 minus
120572)(120572 + 120573) It is easily seen that all these points are in [minus1 1]Therefore
min 119902 (119905) 119905 isin [minus1 1] = min119902 (minus1) 119902 ( minus1
radic6
)
119902 (1
radic6
) 119902 (1) 119902 (120573 minus 120572
120572 + 120573)
(29)
For 119905 = minus1 or 119905 = 1 the functions given by (21) have theform
119891 (119911) = 119911
(1 + 119911)2+ 120573
119911
(1 minus 119911)2 + 120573 = 1 (30)
One can show directly from formula (30) that
1198672(3) = minus1 + 64 (1 minus ) ge minus1 (31)
For 119905 = (120573 minus 120572)(120572 + 120573) there is
1198672(3) = minus1 +
64120572120573 [4 (120572 + 120573) 120572120573 + (120573 minus 120572)2
]
(120572 + 120573)3
(32)
hence
1198672(3) ge minus1 (33)
If 119905 = minus1radic6 or 119905 = 1radic6 then1198672(3) is equal to
1198672(3)
= minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)2
+ 64120572120573
+ (1 minus 120572 minus 120573) [minus10
3(120572 + 120573) minus
64
3radic6
(120572 minus 120573)]
(34)
or
1198672(3)
= minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)2
+ 64120572120573
+ (1 minus 120572 minus 120573) [minus10
3(120572 + 120573) +
64
3radic6
(120572 minus 120573)]
(35)
Abstract and Applied Analysis 5
respectively Without loss of generality we can assume that120572 ge 120573 Then while looking for the minimum value of119867
2(3)
we can restrict the research to the first stated above case (sinceexpression (35) is not less than expression (34))
Transforming (35) we obtain
1198672(3) = minus1 + 64120572120573 minus
4
3(120572 + 120573) (1 minus 120572 minus 120573)
+64
3radic6
(120573 minus 120572) (1 minus 120572 minus 120573)
(36)
Taking the smallest possible 120573 (ie 120573 = 0) the second andthe forth component of this expression will not increase Thevalue of the third component does not depend only on 120573 infact it depends on 120572 + 120573 For this reason we can take 120573 = 0Combining these facts it yields that
1198672(3) ge minus1 minus
4
3120572 (1 minus 120572) minus
64
3radic6
120572 (1 minus 120572) (37)
The smallest value of the right hand side of this inequality isachieved for 120572 = 12 In this case
1198672(3) ge minus1 minus
1
3minus
16
3radic6
= minus4
3minus8
9
radic6 = minus351 (38)
Combining two parts of the proof we obtain the conclu-sion of the theorem Furthermore the above shows that theextremal functions are
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 21199111199050+ 1199112
]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 + 21199111199050+ 1199112
]
(39)
where 1199050= 1radic6
5 Bounds of 1198672(119899) 119899 ge 4 in 119879
It is easily seen that 1198672(119899) le 119899(119899 + 2) for any typically real
function ByTheorem 9 this estimate is sharp providing that119899 is an odd integer At the beginning of this section we willprove the following
Theorem 13 For 119879 one has
min 119886119899119886119899+2
119891 isin 119879 = minus1 (40)
Proof Thecoefficients of the series expansion of function119891 isin
119879 can be written as follows
119886119899= int
120587
0
sin (119899120579)sin 120579
119889] (120579) ] isin 119875[0120587]
(41)
Hence
119886119899119886119899+2
= int
120587
0
sin ((119899 + 1) 120579 minus 120579)
sin 120579119889] (120579)
sdot int
120587
0
sin ((119899 + 1) 120579 + 120579)
sin 120579119889] (120579)
= (int
120587
0
sin (119899 + 1) 120579
sin 120579cos 120579 119889] (120579))
2
minus (int
120587
0
cos (119899 + 1) 120579 119889] (120579))2
(42)
Since
int
120587
0
cos (119899 + 1) 120579 119889] (120579) le int
120587
0
119889] (120579) = 1 (43)
we obtain
119886119899119886119899+2
ge minus1 (44)
In order to prove that the estimate is sharp let us take themeasure ] for which support satisfies condition (119899 + 1)120579 =
120587 This measure corresponds to the function 119891(119911) = 119911(1 minus
2119911cos(120587(119899 + 1)) + 1199112)
Observe that 119886119899119886119899+2
= minus1 holds not only for the measurestated above Namely the value minus1 in (42) is taken also if (119899 +1)120579 = 119896120587 where 119896 is any positive integer less than or equal to119899 From this we conclude that the support of the measure has119899 points 120579
119896= 119896120587(119899 + 1) with weights 120572
119896 119896 = 1 2 119899 such
that sum119899119896=1
120572119896= 1
The weights satisfy
(
119899
sum
119896=1
120572119896(minus1)119896)
2
= 1 (45)
Indeed if the support of ] consists of 119899 points then 119891 takesthe form
119891 (119911) =
119899
sum
119896=1
120572119896
119911
1 minus 2119911 cos 120579119896+ 1199112
(46)
Using trigonometric identities we obtain
119886119899=
119899
sum
119896=1
120572119896(minus1)119896+1
119886119899+2
=
119899
sum
119896=1
120572119896(minus1)119896
(47)
which results in (45)Connecting (45) andsum119899
119896=1120572119896= 1we conclude that119891 is of
the form
119891 (119911) =
119899
sum
119896=1119896 is odd120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112(48)
6 Abstract and Applied Analysis
Table 1 The bounds of the Hankel determinants for functions defined by (51)
Bounds of1198672(119899) Equality in the lower bound for Equality in the upper bound for
minus9 le 1198672(2) le 1 119905 = 1 119905 = 0
minus351 le 1198672(3) le 15 119905 = 0 40 119905 = 1
minus25 le 1198672(4) le 446 119905 = 1 119905 = 061
minus784 le 1198672(5) le 35 119905 = 0 72 119905 = 1
minus49 le 1198672(6) le 967 119905 = 1 119905 = 079
or
119891 (119911) =
119899
sum
119896=1119896 is even120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112 (49)
Itmeans that for even 119899 the support of ] consists of 1198992 pointsand for even 119899 the number of points of the support of ] is equalto (119899 + 1)2 or (119899 minus 1)2
Taking into account |119886119899+1
| le 119899 + 1 and Theorem 13 weobtain the following
Theorem 14 For 119879 one has
1198672(119899) ge minus (119899 + 1)
2minus 1 (50)
Unfortunately this bound is not sharp However thefollowing can be conjectured
Conjecture 15 For any positive integer 119899 the following esti-mate 119867
2(119899) ge minus(119899 + 1)
2 holds Moreover this bound is sharpfor even 119899
This conjecture is supported by the facts that in thetheorems concerning119867
2(2) and119867
2(3) the extremal functions
are of the form
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 2119911119905 + 1199112]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 minus 2119911119905 + 1199112]
(51)
for appropriately taken 119905 isin [minus1 1] The exact bounds ofthe Hankel determinants for these functions are collected inTable 1 They were obtained numerically
6 Remarks Concerning 1198672(119899) in 119879
(2)
In class 119879 we discuss subclass 119879(2) consisting of the functionswhich are odd The definition of this class is
119879(2)
= 119891 isin 119879 119891 (minus119911) = minus119891 (119911) 119911 isin Δ (52)
For 119891 isin 119879(2) the representation formula similar to (2) is
valid Namely
119891 (119911) = int
1
minus1
119911 (1 + 1199112)
(1 + 1199112)2
minus 411991121199052119889] (119905) ] isin 119875
[minus11] (53)
Function 119891 has the Taylor series expansion
119891 (119911) = sum
119899 is odd119886119899119911119899 119886119899= int
1
minus1
119880119899minus1
(119905) 119889] (119905) (54)
The following inequalities are obvious
minus (119899 + 1)2le 1198672(119899) le 0 for even 119899
1198672(119899) le (119899 + 1)
2minus 1 for odd 119899
(55)
equalities hold for 119891(119911) = 119911(1 + 1199112)(1 minus 119911
2)2
For a given class 119860 sub A let us denote by Ψ119899(119860) 119899 ge 1
the set (119886119899 119886119899+2
) 119891 isin 119860 From (53) it follows that Ψ119899(119879) is
the closed convex hull of the curve
120582 [minus1 1] ni 119905 997888rarr (119880119899minus1
(119905) 119880119899+1
(119905)) (56)
FromTheorem 13 and from the equivalence
(119886119899 119886119899+2
) isin Ψ119899(119879) lArrrArr (119886
119899 119886119899+2
) isin Ψ119899(119879(2)) (57)
we get
min 119886119899119886119899+2
119891 isin 119879(2) = minus1 (58)
Hence for odd 119899 we know that
min 1198672(119899) 119891 isin 119879
(2) ge minus1 (59)
The equality holds for functions (48) or (49) providing that120572119896
= 120572119899+1minus119896
Then connecting the components of theseformulae in pairs we obtain
120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112
+ 120572119899+1minus119896
119911
1 minus 2119911 cos ((119899 + 1 minus 119896) 120587 (119899 + 1)) + 1199112
= 2120572119896
119911 (1 + 1199112)
(1 + 1199112)2
minus 41199112cos2 (119896120587 (119899 + 1))
(60)
With help of the argument given in the proof of Theorem 13we eventually obtain the odd functions for which 119886
119899119886119899+2
=
minus1
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 7
References
[1] C Pommerenke ldquoOn the coefficients and Hankel determinantsof univalent functionsrdquo Journal of the London MathematicalSociety vol 41 pp 111ndash122 1966
[2] C Pommerenke ldquoOn the Hankel determinants of univalentfunctionsrdquoMathematika vol 14 pp 108ndash112 1967
[3] A Janteng S A Halim and M Darus ldquoCoefficient inequalityfor a function whose derivative has a positive real partrdquo Journalof Inequalities in Pure and Applied Mathematics vol 7 no 2article 50 2006
[4] A Janteng S A Halim andMDarus ldquoHankel determinant forstarlike and convex functionsrdquo International Journal of Mathe-matical Analysis vol 1 no 13 pp 619ndash625 2007
[5] S K Lee V Ravichandran and S Supramaniam ldquoBounds forthe second Hankel determinant of certain univalent functionsrdquoJournal of Inequalities and Applications vol 2013 article 2812013
[6] D Vamshee Krishna andT Ramreddy ldquoHankel determinant forstarlike and convex functions of order alphardquo Tbilisi Mathemat-ical Journal vol 5 no 1 pp 65ndash76 2012
[7] C Selvaraj and T R K Kumar ldquoSecond Hankel determinantfor certain classes of analytic functionsrdquo International Journalof Applied Mathematics vol 28 no 1 pp 37ndash50 2015
[8] W Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis Conference Proceedings and LectureNotes in Analysis Vol 1 pp 157ndash169 International PressTianjin China 1992
[9] J Sokoł and J Stankiewicz ldquoRadius of convexity of some sub-classes of strongly starlike functionsrdquoZeszytyNaukowe Politech-niki Rzeszowskiej Matematyka vol 19 pp 101ndash105 1996
[10] M-S Liu J-F Xu and M Yang ldquoUpper bound of second Han-kel determinant for certain subclasses of analytic functionsrdquoAbstract and Applied Analysis vol 2014 Article ID 603180 10pages 2014
[11] A K Mishra and P Gochhayat ldquoSecond Hankel determinantfor a class of analytic functions defined by fractional deriva-tiverdquo International Journal of Mathematics and MathematicalSciences vol 2008 Article ID 153280 10 pages 2008
[12] J W Noonan and D K Thomas ldquoOn the Hankel determinantsof areally mean 119901-valent functionsrdquo Proceedings of the LondonMathematical Society vol 25 pp 503ndash524 1972
[13] G Singh ldquoHankel determinant for a new subclass of analyticfunctionsrdquo Scientia Magna vol 8 no 4 pp 61ndash65 2012
[14] G Singh ldquoHankel determinant for analytic functions withrespect to other pointsrdquo Engineering Mathematics Letters vol2 no 2 pp 115ndash123 2013
[15] T Hayami and S Owa ldquoGeneralized hankel determinant forcertain classesrdquo International Journal of Mathematical Analysisvol 4 no 52 pp 2573ndash2585 2010
[16] K O Babalola ldquoOnH3(1) Hankel determinants for some classes
of univalent functionsrdquo in Inequality Theory and Applicationsvol 6 17 Nova Science Publishers New York NY USA 2010
[17] G Shanmugam B Adolf Stephen and K O Babalola ldquoThirdHankel determinant for 120572-starlike functionsrdquo Gulf Journal ofMathematics vol 2 no 2 pp 107ndash113 2014
[18] P T Mocanu ldquoUne proprit de convexit gnralise dans la thoriede la representation conformerdquoMathematica (Cluj) vol 11 no34 pp 127ndash133 1969
[19] M S Robertson ldquoOn the coefficients of a typically-real func-tionrdquo Bulletin of the American Mathematical Society vol 41 no8 pp 565ndash572 1935
[20] D J Hallenbeck and T H MacGregor Linear Problems andConvexity Techniques in Geometric Function Theory Mono-graphs and Studies in Mathematics 22 Pitman AdavancedPublishing Program Boston Mass USA 1984
[21] W Ma ldquoGeneralized Zalcman conjecture for starlike and typ-ically real functionsrdquo Journal of Mathematical Analysis andApplications vol 234 no 1 pp 328ndash339 1999
[22] W Szapiel Extremal problems for convex sets Applications toholomorphic functions [Dissertation XXXVII] UMCS PressLublin Poland reviewer J Krzyz 1986 (Polish)
[23] W Szapiel ldquoExtreme points of convex sets (II) Inuence of nor-malization on integral representationrdquo Bulletin de lrsquoAcademiePolonaise des Sciences Serie des SciencesMathematiques vol 29pp 535ndash544 1981
Submit your manuscripts athttpwwwhindawicom
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Corollary 10 For 119879 one has
max 1198672(3) 119891 isin 119879 = 15 (19)
In similar way as it was done for Lemma 7 one can provethe following
Lemma 11 The boundary of Ω3(119879) consists of points (119886
3
1198864 1198865) that correspond to the following functions
119891 (119911) = 120572119911
1 minus 21199111199051+ 1199112
+ (1 minus 120572)119911
1 minus 21199111199052+ 1199112
120572 isin [0 1] 1199051 1199052 isin [minus1 1]
(20)
or
119891 (119911) = 120572119911
(1 + 119911)2+ 120573
119911
(1 minus 119911)2
+ (1 minus 120572 minus 120573)119911
1 minus 2119911119905 + 1199112
120572 120573 isin [0 1] 120572 + 120573 le 1 119905 isin [minus1 1]
(21)
Theorem 12 For 119879 one has
min 1198672(3) 119891 isin 119879 = minus
4
3+8
9
radic6 = minus351 (22)
Proof By Lemma 11 it suffices to discuss functions given by(20) or (21) Consider the following
(I) For functions (20) we have
1198863= (41199051
2minus 1) 120572 + (4119905
2
2minus 1) (1 minus 120572)
1198864= (81199051
3minus 41199051) 120572 + (8119905
2
3minus 41199052) (1 minus 120572)
1198865= (16119905
1
4minus 121199051
2+ 1) 120572
+ (161199052
4minus 121199052
2+ 1) (1 minus 120572)
(23)
and hence applying the Turan identity1198672(3) = 119892
3(120572 1199051 1199052)
where
1198923(120572 1199051 1199052)
= minus1205722minus (1 minus 120572)
2
+ 2120572 (1 minus 120572) [8 (1199051minus 1199052)2
(1 minus 1199051
2) (1 minus 119905
2
2) minus 1]
120572 isin [0 1] 1199051 1199052 isin [minus1 1]
(24)
The expression in brackets is greater than or equal to minus1for all 119905
1 1199052isin [minus1 1] Hence
1198672(3) ge minus120572
2minus (1 minus 120572)
2minus 2120572 (1 minus 120572) = minus1 (25)
(II) If function 119891 is given by (21) then
1198863= 3 (120572 + 120573) + (4119905
2minus 1) (1 minus 120572 minus 120573)
1198864= 4 (120573 minus 120572) + (8119905
3minus 4119905) (1 minus 120572 minus 120573)
1198865= 5 (120572 + 120573) + (16119905
4minus 121199052+ 1) (1 minus 120572 minus 120573)
(26)
Using the Turan identity it follows that 1198672(3) = 119892
4(120572 120573 119905)
where
1198924(120572 120573 119905) = minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)
2
+ 64120572120573
+ 2 (1 minus 120572 minus 120573) 119902 (119905)
119902 (119905) = (120572 + 120573) (241199054minus 81199052minus 1)
+ (120572 minus 120573) (321199053minus 16119905)
(27)
under the assumptions 120572 120573 isin [0 1] 120572+120573 le 1 and 119905 isin [minus1 1]Let 120572 and 120573 be fixed Since
120597119902
120597119905= 8 (6119905
2minus 1) [119905 (120572 + 120573) + 120572 minus 120573] (28)
the critical points of 119902 are as follows minus1radic6 1radic6 and (120573 minus
120572)(120572 + 120573) It is easily seen that all these points are in [minus1 1]Therefore
min 119902 (119905) 119905 isin [minus1 1] = min119902 (minus1) 119902 ( minus1
radic6
)
119902 (1
radic6
) 119902 (1) 119902 (120573 minus 120572
120572 + 120573)
(29)
For 119905 = minus1 or 119905 = 1 the functions given by (21) have theform
119891 (119911) = 119911
(1 + 119911)2+ 120573
119911
(1 minus 119911)2 + 120573 = 1 (30)
One can show directly from formula (30) that
1198672(3) = minus1 + 64 (1 minus ) ge minus1 (31)
For 119905 = (120573 minus 120572)(120572 + 120573) there is
1198672(3) = minus1 +
64120572120573 [4 (120572 + 120573) 120572120573 + (120573 minus 120572)2
]
(120572 + 120573)3
(32)
hence
1198672(3) ge minus1 (33)
If 119905 = minus1radic6 or 119905 = 1radic6 then1198672(3) is equal to
1198672(3)
= minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)2
+ 64120572120573
+ (1 minus 120572 minus 120573) [minus10
3(120572 + 120573) minus
64
3radic6
(120572 minus 120573)]
(34)
or
1198672(3)
= minus1 + 2 (120572 + 120573) minus 2 (120572 + 120573)2
+ 64120572120573
+ (1 minus 120572 minus 120573) [minus10
3(120572 + 120573) +
64
3radic6
(120572 minus 120573)]
(35)
Abstract and Applied Analysis 5
respectively Without loss of generality we can assume that120572 ge 120573 Then while looking for the minimum value of119867
2(3)
we can restrict the research to the first stated above case (sinceexpression (35) is not less than expression (34))
Transforming (35) we obtain
1198672(3) = minus1 + 64120572120573 minus
4
3(120572 + 120573) (1 minus 120572 minus 120573)
+64
3radic6
(120573 minus 120572) (1 minus 120572 minus 120573)
(36)
Taking the smallest possible 120573 (ie 120573 = 0) the second andthe forth component of this expression will not increase Thevalue of the third component does not depend only on 120573 infact it depends on 120572 + 120573 For this reason we can take 120573 = 0Combining these facts it yields that
1198672(3) ge minus1 minus
4
3120572 (1 minus 120572) minus
64
3radic6
120572 (1 minus 120572) (37)
The smallest value of the right hand side of this inequality isachieved for 120572 = 12 In this case
1198672(3) ge minus1 minus
1
3minus
16
3radic6
= minus4
3minus8
9
radic6 = minus351 (38)
Combining two parts of the proof we obtain the conclu-sion of the theorem Furthermore the above shows that theextremal functions are
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 21199111199050+ 1199112
]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 + 21199111199050+ 1199112
]
(39)
where 1199050= 1radic6
5 Bounds of 1198672(119899) 119899 ge 4 in 119879
It is easily seen that 1198672(119899) le 119899(119899 + 2) for any typically real
function ByTheorem 9 this estimate is sharp providing that119899 is an odd integer At the beginning of this section we willprove the following
Theorem 13 For 119879 one has
min 119886119899119886119899+2
119891 isin 119879 = minus1 (40)
Proof Thecoefficients of the series expansion of function119891 isin
119879 can be written as follows
119886119899= int
120587
0
sin (119899120579)sin 120579
119889] (120579) ] isin 119875[0120587]
(41)
Hence
119886119899119886119899+2
= int
120587
0
sin ((119899 + 1) 120579 minus 120579)
sin 120579119889] (120579)
sdot int
120587
0
sin ((119899 + 1) 120579 + 120579)
sin 120579119889] (120579)
= (int
120587
0
sin (119899 + 1) 120579
sin 120579cos 120579 119889] (120579))
2
minus (int
120587
0
cos (119899 + 1) 120579 119889] (120579))2
(42)
Since
int
120587
0
cos (119899 + 1) 120579 119889] (120579) le int
120587
0
119889] (120579) = 1 (43)
we obtain
119886119899119886119899+2
ge minus1 (44)
In order to prove that the estimate is sharp let us take themeasure ] for which support satisfies condition (119899 + 1)120579 =
120587 This measure corresponds to the function 119891(119911) = 119911(1 minus
2119911cos(120587(119899 + 1)) + 1199112)
Observe that 119886119899119886119899+2
= minus1 holds not only for the measurestated above Namely the value minus1 in (42) is taken also if (119899 +1)120579 = 119896120587 where 119896 is any positive integer less than or equal to119899 From this we conclude that the support of the measure has119899 points 120579
119896= 119896120587(119899 + 1) with weights 120572
119896 119896 = 1 2 119899 such
that sum119899119896=1
120572119896= 1
The weights satisfy
(
119899
sum
119896=1
120572119896(minus1)119896)
2
= 1 (45)
Indeed if the support of ] consists of 119899 points then 119891 takesthe form
119891 (119911) =
119899
sum
119896=1
120572119896
119911
1 minus 2119911 cos 120579119896+ 1199112
(46)
Using trigonometric identities we obtain
119886119899=
119899
sum
119896=1
120572119896(minus1)119896+1
119886119899+2
=
119899
sum
119896=1
120572119896(minus1)119896
(47)
which results in (45)Connecting (45) andsum119899
119896=1120572119896= 1we conclude that119891 is of
the form
119891 (119911) =
119899
sum
119896=1119896 is odd120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112(48)
6 Abstract and Applied Analysis
Table 1 The bounds of the Hankel determinants for functions defined by (51)
Bounds of1198672(119899) Equality in the lower bound for Equality in the upper bound for
minus9 le 1198672(2) le 1 119905 = 1 119905 = 0
minus351 le 1198672(3) le 15 119905 = 0 40 119905 = 1
minus25 le 1198672(4) le 446 119905 = 1 119905 = 061
minus784 le 1198672(5) le 35 119905 = 0 72 119905 = 1
minus49 le 1198672(6) le 967 119905 = 1 119905 = 079
or
119891 (119911) =
119899
sum
119896=1119896 is even120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112 (49)
Itmeans that for even 119899 the support of ] consists of 1198992 pointsand for even 119899 the number of points of the support of ] is equalto (119899 + 1)2 or (119899 minus 1)2
Taking into account |119886119899+1
| le 119899 + 1 and Theorem 13 weobtain the following
Theorem 14 For 119879 one has
1198672(119899) ge minus (119899 + 1)
2minus 1 (50)
Unfortunately this bound is not sharp However thefollowing can be conjectured
Conjecture 15 For any positive integer 119899 the following esti-mate 119867
2(119899) ge minus(119899 + 1)
2 holds Moreover this bound is sharpfor even 119899
This conjecture is supported by the facts that in thetheorems concerning119867
2(2) and119867
2(3) the extremal functions
are of the form
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 2119911119905 + 1199112]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 minus 2119911119905 + 1199112]
(51)
for appropriately taken 119905 isin [minus1 1] The exact bounds ofthe Hankel determinants for these functions are collected inTable 1 They were obtained numerically
6 Remarks Concerning 1198672(119899) in 119879
(2)
In class 119879 we discuss subclass 119879(2) consisting of the functionswhich are odd The definition of this class is
119879(2)
= 119891 isin 119879 119891 (minus119911) = minus119891 (119911) 119911 isin Δ (52)
For 119891 isin 119879(2) the representation formula similar to (2) is
valid Namely
119891 (119911) = int
1
minus1
119911 (1 + 1199112)
(1 + 1199112)2
minus 411991121199052119889] (119905) ] isin 119875
[minus11] (53)
Function 119891 has the Taylor series expansion
119891 (119911) = sum
119899 is odd119886119899119911119899 119886119899= int
1
minus1
119880119899minus1
(119905) 119889] (119905) (54)
The following inequalities are obvious
minus (119899 + 1)2le 1198672(119899) le 0 for even 119899
1198672(119899) le (119899 + 1)
2minus 1 for odd 119899
(55)
equalities hold for 119891(119911) = 119911(1 + 1199112)(1 minus 119911
2)2
For a given class 119860 sub A let us denote by Ψ119899(119860) 119899 ge 1
the set (119886119899 119886119899+2
) 119891 isin 119860 From (53) it follows that Ψ119899(119879) is
the closed convex hull of the curve
120582 [minus1 1] ni 119905 997888rarr (119880119899minus1
(119905) 119880119899+1
(119905)) (56)
FromTheorem 13 and from the equivalence
(119886119899 119886119899+2
) isin Ψ119899(119879) lArrrArr (119886
119899 119886119899+2
) isin Ψ119899(119879(2)) (57)
we get
min 119886119899119886119899+2
119891 isin 119879(2) = minus1 (58)
Hence for odd 119899 we know that
min 1198672(119899) 119891 isin 119879
(2) ge minus1 (59)
The equality holds for functions (48) or (49) providing that120572119896
= 120572119899+1minus119896
Then connecting the components of theseformulae in pairs we obtain
120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112
+ 120572119899+1minus119896
119911
1 minus 2119911 cos ((119899 + 1 minus 119896) 120587 (119899 + 1)) + 1199112
= 2120572119896
119911 (1 + 1199112)
(1 + 1199112)2
minus 41199112cos2 (119896120587 (119899 + 1))
(60)
With help of the argument given in the proof of Theorem 13we eventually obtain the odd functions for which 119886
119899119886119899+2
=
minus1
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 7
References
[1] C Pommerenke ldquoOn the coefficients and Hankel determinantsof univalent functionsrdquo Journal of the London MathematicalSociety vol 41 pp 111ndash122 1966
[2] C Pommerenke ldquoOn the Hankel determinants of univalentfunctionsrdquoMathematika vol 14 pp 108ndash112 1967
[3] A Janteng S A Halim and M Darus ldquoCoefficient inequalityfor a function whose derivative has a positive real partrdquo Journalof Inequalities in Pure and Applied Mathematics vol 7 no 2article 50 2006
[4] A Janteng S A Halim andMDarus ldquoHankel determinant forstarlike and convex functionsrdquo International Journal of Mathe-matical Analysis vol 1 no 13 pp 619ndash625 2007
[5] S K Lee V Ravichandran and S Supramaniam ldquoBounds forthe second Hankel determinant of certain univalent functionsrdquoJournal of Inequalities and Applications vol 2013 article 2812013
[6] D Vamshee Krishna andT Ramreddy ldquoHankel determinant forstarlike and convex functions of order alphardquo Tbilisi Mathemat-ical Journal vol 5 no 1 pp 65ndash76 2012
[7] C Selvaraj and T R K Kumar ldquoSecond Hankel determinantfor certain classes of analytic functionsrdquo International Journalof Applied Mathematics vol 28 no 1 pp 37ndash50 2015
[8] W Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis Conference Proceedings and LectureNotes in Analysis Vol 1 pp 157ndash169 International PressTianjin China 1992
[9] J Sokoł and J Stankiewicz ldquoRadius of convexity of some sub-classes of strongly starlike functionsrdquoZeszytyNaukowe Politech-niki Rzeszowskiej Matematyka vol 19 pp 101ndash105 1996
[10] M-S Liu J-F Xu and M Yang ldquoUpper bound of second Han-kel determinant for certain subclasses of analytic functionsrdquoAbstract and Applied Analysis vol 2014 Article ID 603180 10pages 2014
[11] A K Mishra and P Gochhayat ldquoSecond Hankel determinantfor a class of analytic functions defined by fractional deriva-tiverdquo International Journal of Mathematics and MathematicalSciences vol 2008 Article ID 153280 10 pages 2008
[12] J W Noonan and D K Thomas ldquoOn the Hankel determinantsof areally mean 119901-valent functionsrdquo Proceedings of the LondonMathematical Society vol 25 pp 503ndash524 1972
[13] G Singh ldquoHankel determinant for a new subclass of analyticfunctionsrdquo Scientia Magna vol 8 no 4 pp 61ndash65 2012
[14] G Singh ldquoHankel determinant for analytic functions withrespect to other pointsrdquo Engineering Mathematics Letters vol2 no 2 pp 115ndash123 2013
[15] T Hayami and S Owa ldquoGeneralized hankel determinant forcertain classesrdquo International Journal of Mathematical Analysisvol 4 no 52 pp 2573ndash2585 2010
[16] K O Babalola ldquoOnH3(1) Hankel determinants for some classes
of univalent functionsrdquo in Inequality Theory and Applicationsvol 6 17 Nova Science Publishers New York NY USA 2010
[17] G Shanmugam B Adolf Stephen and K O Babalola ldquoThirdHankel determinant for 120572-starlike functionsrdquo Gulf Journal ofMathematics vol 2 no 2 pp 107ndash113 2014
[18] P T Mocanu ldquoUne proprit de convexit gnralise dans la thoriede la representation conformerdquoMathematica (Cluj) vol 11 no34 pp 127ndash133 1969
[19] M S Robertson ldquoOn the coefficients of a typically-real func-tionrdquo Bulletin of the American Mathematical Society vol 41 no8 pp 565ndash572 1935
[20] D J Hallenbeck and T H MacGregor Linear Problems andConvexity Techniques in Geometric Function Theory Mono-graphs and Studies in Mathematics 22 Pitman AdavancedPublishing Program Boston Mass USA 1984
[21] W Ma ldquoGeneralized Zalcman conjecture for starlike and typ-ically real functionsrdquo Journal of Mathematical Analysis andApplications vol 234 no 1 pp 328ndash339 1999
[22] W Szapiel Extremal problems for convex sets Applications toholomorphic functions [Dissertation XXXVII] UMCS PressLublin Poland reviewer J Krzyz 1986 (Polish)
[23] W Szapiel ldquoExtreme points of convex sets (II) Inuence of nor-malization on integral representationrdquo Bulletin de lrsquoAcademiePolonaise des Sciences Serie des SciencesMathematiques vol 29pp 535ndash544 1981
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
Abstract and Applied Analysis 5
respectively Without loss of generality we can assume that120572 ge 120573 Then while looking for the minimum value of119867
2(3)
we can restrict the research to the first stated above case (sinceexpression (35) is not less than expression (34))
Transforming (35) we obtain
1198672(3) = minus1 + 64120572120573 minus
4
3(120572 + 120573) (1 minus 120572 minus 120573)
+64
3radic6
(120573 minus 120572) (1 minus 120572 minus 120573)
(36)
Taking the smallest possible 120573 (ie 120573 = 0) the second andthe forth component of this expression will not increase Thevalue of the third component does not depend only on 120573 infact it depends on 120572 + 120573 For this reason we can take 120573 = 0Combining these facts it yields that
1198672(3) ge minus1 minus
4
3120572 (1 minus 120572) minus
64
3radic6
120572 (1 minus 120572) (37)
The smallest value of the right hand side of this inequality isachieved for 120572 = 12 In this case
1198672(3) ge minus1 minus
1
3minus
16
3radic6
= minus4
3minus8
9
radic6 = minus351 (38)
Combining two parts of the proof we obtain the conclu-sion of the theorem Furthermore the above shows that theextremal functions are
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 21199111199050+ 1199112
]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 + 21199111199050+ 1199112
]
(39)
where 1199050= 1radic6
5 Bounds of 1198672(119899) 119899 ge 4 in 119879
It is easily seen that 1198672(119899) le 119899(119899 + 2) for any typically real
function ByTheorem 9 this estimate is sharp providing that119899 is an odd integer At the beginning of this section we willprove the following
Theorem 13 For 119879 one has
min 119886119899119886119899+2
119891 isin 119879 = minus1 (40)
Proof Thecoefficients of the series expansion of function119891 isin
119879 can be written as follows
119886119899= int
120587
0
sin (119899120579)sin 120579
119889] (120579) ] isin 119875[0120587]
(41)
Hence
119886119899119886119899+2
= int
120587
0
sin ((119899 + 1) 120579 minus 120579)
sin 120579119889] (120579)
sdot int
120587
0
sin ((119899 + 1) 120579 + 120579)
sin 120579119889] (120579)
= (int
120587
0
sin (119899 + 1) 120579
sin 120579cos 120579 119889] (120579))
2
minus (int
120587
0
cos (119899 + 1) 120579 119889] (120579))2
(42)
Since
int
120587
0
cos (119899 + 1) 120579 119889] (120579) le int
120587
0
119889] (120579) = 1 (43)
we obtain
119886119899119886119899+2
ge minus1 (44)
In order to prove that the estimate is sharp let us take themeasure ] for which support satisfies condition (119899 + 1)120579 =
120587 This measure corresponds to the function 119891(119911) = 119911(1 minus
2119911cos(120587(119899 + 1)) + 1199112)
Observe that 119886119899119886119899+2
= minus1 holds not only for the measurestated above Namely the value minus1 in (42) is taken also if (119899 +1)120579 = 119896120587 where 119896 is any positive integer less than or equal to119899 From this we conclude that the support of the measure has119899 points 120579
119896= 119896120587(119899 + 1) with weights 120572
119896 119896 = 1 2 119899 such
that sum119899119896=1
120572119896= 1
The weights satisfy
(
119899
sum
119896=1
120572119896(minus1)119896)
2
= 1 (45)
Indeed if the support of ] consists of 119899 points then 119891 takesthe form
119891 (119911) =
119899
sum
119896=1
120572119896
119911
1 minus 2119911 cos 120579119896+ 1199112
(46)
Using trigonometric identities we obtain
119886119899=
119899
sum
119896=1
120572119896(minus1)119896+1
119886119899+2
=
119899
sum
119896=1
120572119896(minus1)119896
(47)
which results in (45)Connecting (45) andsum119899
119896=1120572119896= 1we conclude that119891 is of
the form
119891 (119911) =
119899
sum
119896=1119896 is odd120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112(48)
6 Abstract and Applied Analysis
Table 1 The bounds of the Hankel determinants for functions defined by (51)
Bounds of1198672(119899) Equality in the lower bound for Equality in the upper bound for
minus9 le 1198672(2) le 1 119905 = 1 119905 = 0
minus351 le 1198672(3) le 15 119905 = 0 40 119905 = 1
minus25 le 1198672(4) le 446 119905 = 1 119905 = 061
minus784 le 1198672(5) le 35 119905 = 0 72 119905 = 1
minus49 le 1198672(6) le 967 119905 = 1 119905 = 079
or
119891 (119911) =
119899
sum
119896=1119896 is even120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112 (49)
Itmeans that for even 119899 the support of ] consists of 1198992 pointsand for even 119899 the number of points of the support of ] is equalto (119899 + 1)2 or (119899 minus 1)2
Taking into account |119886119899+1
| le 119899 + 1 and Theorem 13 weobtain the following
Theorem 14 For 119879 one has
1198672(119899) ge minus (119899 + 1)
2minus 1 (50)
Unfortunately this bound is not sharp However thefollowing can be conjectured
Conjecture 15 For any positive integer 119899 the following esti-mate 119867
2(119899) ge minus(119899 + 1)
2 holds Moreover this bound is sharpfor even 119899
This conjecture is supported by the facts that in thetheorems concerning119867
2(2) and119867
2(3) the extremal functions
are of the form
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 2119911119905 + 1199112]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 minus 2119911119905 + 1199112]
(51)
for appropriately taken 119905 isin [minus1 1] The exact bounds ofthe Hankel determinants for these functions are collected inTable 1 They were obtained numerically
6 Remarks Concerning 1198672(119899) in 119879
(2)
In class 119879 we discuss subclass 119879(2) consisting of the functionswhich are odd The definition of this class is
119879(2)
= 119891 isin 119879 119891 (minus119911) = minus119891 (119911) 119911 isin Δ (52)
For 119891 isin 119879(2) the representation formula similar to (2) is
valid Namely
119891 (119911) = int
1
minus1
119911 (1 + 1199112)
(1 + 1199112)2
minus 411991121199052119889] (119905) ] isin 119875
[minus11] (53)
Function 119891 has the Taylor series expansion
119891 (119911) = sum
119899 is odd119886119899119911119899 119886119899= int
1
minus1
119880119899minus1
(119905) 119889] (119905) (54)
The following inequalities are obvious
minus (119899 + 1)2le 1198672(119899) le 0 for even 119899
1198672(119899) le (119899 + 1)
2minus 1 for odd 119899
(55)
equalities hold for 119891(119911) = 119911(1 + 1199112)(1 minus 119911
2)2
For a given class 119860 sub A let us denote by Ψ119899(119860) 119899 ge 1
the set (119886119899 119886119899+2
) 119891 isin 119860 From (53) it follows that Ψ119899(119879) is
the closed convex hull of the curve
120582 [minus1 1] ni 119905 997888rarr (119880119899minus1
(119905) 119880119899+1
(119905)) (56)
FromTheorem 13 and from the equivalence
(119886119899 119886119899+2
) isin Ψ119899(119879) lArrrArr (119886
119899 119886119899+2
) isin Ψ119899(119879(2)) (57)
we get
min 119886119899119886119899+2
119891 isin 119879(2) = minus1 (58)
Hence for odd 119899 we know that
min 1198672(119899) 119891 isin 119879
(2) ge minus1 (59)
The equality holds for functions (48) or (49) providing that120572119896
= 120572119899+1minus119896
Then connecting the components of theseformulae in pairs we obtain
120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112
+ 120572119899+1minus119896
119911
1 minus 2119911 cos ((119899 + 1 minus 119896) 120587 (119899 + 1)) + 1199112
= 2120572119896
119911 (1 + 1199112)
(1 + 1199112)2
minus 41199112cos2 (119896120587 (119899 + 1))
(60)
With help of the argument given in the proof of Theorem 13we eventually obtain the odd functions for which 119886
119899119886119899+2
=
minus1
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 7
References
[1] C Pommerenke ldquoOn the coefficients and Hankel determinantsof univalent functionsrdquo Journal of the London MathematicalSociety vol 41 pp 111ndash122 1966
[2] C Pommerenke ldquoOn the Hankel determinants of univalentfunctionsrdquoMathematika vol 14 pp 108ndash112 1967
[3] A Janteng S A Halim and M Darus ldquoCoefficient inequalityfor a function whose derivative has a positive real partrdquo Journalof Inequalities in Pure and Applied Mathematics vol 7 no 2article 50 2006
[4] A Janteng S A Halim andMDarus ldquoHankel determinant forstarlike and convex functionsrdquo International Journal of Mathe-matical Analysis vol 1 no 13 pp 619ndash625 2007
[5] S K Lee V Ravichandran and S Supramaniam ldquoBounds forthe second Hankel determinant of certain univalent functionsrdquoJournal of Inequalities and Applications vol 2013 article 2812013
[6] D Vamshee Krishna andT Ramreddy ldquoHankel determinant forstarlike and convex functions of order alphardquo Tbilisi Mathemat-ical Journal vol 5 no 1 pp 65ndash76 2012
[7] C Selvaraj and T R K Kumar ldquoSecond Hankel determinantfor certain classes of analytic functionsrdquo International Journalof Applied Mathematics vol 28 no 1 pp 37ndash50 2015
[8] W Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis Conference Proceedings and LectureNotes in Analysis Vol 1 pp 157ndash169 International PressTianjin China 1992
[9] J Sokoł and J Stankiewicz ldquoRadius of convexity of some sub-classes of strongly starlike functionsrdquoZeszytyNaukowe Politech-niki Rzeszowskiej Matematyka vol 19 pp 101ndash105 1996
[10] M-S Liu J-F Xu and M Yang ldquoUpper bound of second Han-kel determinant for certain subclasses of analytic functionsrdquoAbstract and Applied Analysis vol 2014 Article ID 603180 10pages 2014
[11] A K Mishra and P Gochhayat ldquoSecond Hankel determinantfor a class of analytic functions defined by fractional deriva-tiverdquo International Journal of Mathematics and MathematicalSciences vol 2008 Article ID 153280 10 pages 2008
[12] J W Noonan and D K Thomas ldquoOn the Hankel determinantsof areally mean 119901-valent functionsrdquo Proceedings of the LondonMathematical Society vol 25 pp 503ndash524 1972
[13] G Singh ldquoHankel determinant for a new subclass of analyticfunctionsrdquo Scientia Magna vol 8 no 4 pp 61ndash65 2012
[14] G Singh ldquoHankel determinant for analytic functions withrespect to other pointsrdquo Engineering Mathematics Letters vol2 no 2 pp 115ndash123 2013
[15] T Hayami and S Owa ldquoGeneralized hankel determinant forcertain classesrdquo International Journal of Mathematical Analysisvol 4 no 52 pp 2573ndash2585 2010
[16] K O Babalola ldquoOnH3(1) Hankel determinants for some classes
of univalent functionsrdquo in Inequality Theory and Applicationsvol 6 17 Nova Science Publishers New York NY USA 2010
[17] G Shanmugam B Adolf Stephen and K O Babalola ldquoThirdHankel determinant for 120572-starlike functionsrdquo Gulf Journal ofMathematics vol 2 no 2 pp 107ndash113 2014
[18] P T Mocanu ldquoUne proprit de convexit gnralise dans la thoriede la representation conformerdquoMathematica (Cluj) vol 11 no34 pp 127ndash133 1969
[19] M S Robertson ldquoOn the coefficients of a typically-real func-tionrdquo Bulletin of the American Mathematical Society vol 41 no8 pp 565ndash572 1935
[20] D J Hallenbeck and T H MacGregor Linear Problems andConvexity Techniques in Geometric Function Theory Mono-graphs and Studies in Mathematics 22 Pitman AdavancedPublishing Program Boston Mass USA 1984
[21] W Ma ldquoGeneralized Zalcman conjecture for starlike and typ-ically real functionsrdquo Journal of Mathematical Analysis andApplications vol 234 no 1 pp 328ndash339 1999
[22] W Szapiel Extremal problems for convex sets Applications toholomorphic functions [Dissertation XXXVII] UMCS PressLublin Poland reviewer J Krzyz 1986 (Polish)
[23] W Szapiel ldquoExtreme points of convex sets (II) Inuence of nor-malization on integral representationrdquo Bulletin de lrsquoAcademiePolonaise des Sciences Serie des SciencesMathematiques vol 29pp 535ndash544 1981
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
6 Abstract and Applied Analysis
Table 1 The bounds of the Hankel determinants for functions defined by (51)
Bounds of1198672(119899) Equality in the lower bound for Equality in the upper bound for
minus9 le 1198672(2) le 1 119905 = 1 119905 = 0
minus351 le 1198672(3) le 15 119905 = 0 40 119905 = 1
minus25 le 1198672(4) le 446 119905 = 1 119905 = 061
minus784 le 1198672(5) le 35 119905 = 0 72 119905 = 1
minus49 le 1198672(6) le 967 119905 = 1 119905 = 079
or
119891 (119911) =
119899
sum
119896=1119896 is even120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112 (49)
Itmeans that for even 119899 the support of ] consists of 1198992 pointsand for even 119899 the number of points of the support of ] is equalto (119899 + 1)2 or (119899 minus 1)2
Taking into account |119886119899+1
| le 119899 + 1 and Theorem 13 weobtain the following
Theorem 14 For 119879 one has
1198672(119899) ge minus (119899 + 1)
2minus 1 (50)
Unfortunately this bound is not sharp However thefollowing can be conjectured
Conjecture 15 For any positive integer 119899 the following esti-mate 119867
2(119899) ge minus(119899 + 1)
2 holds Moreover this bound is sharpfor even 119899
This conjecture is supported by the facts that in thetheorems concerning119867
2(2) and119867
2(3) the extremal functions
are of the form
119891 (119911) =1
2[
119911
(1 + 119911)2+
119911
1 minus 2119911119905 + 1199112]
119891 (119911) =1
2[
119911
(1 minus 119911)2+
119911
1 minus 2119911119905 + 1199112]
(51)
for appropriately taken 119905 isin [minus1 1] The exact bounds ofthe Hankel determinants for these functions are collected inTable 1 They were obtained numerically
6 Remarks Concerning 1198672(119899) in 119879
(2)
In class 119879 we discuss subclass 119879(2) consisting of the functionswhich are odd The definition of this class is
119879(2)
= 119891 isin 119879 119891 (minus119911) = minus119891 (119911) 119911 isin Δ (52)
For 119891 isin 119879(2) the representation formula similar to (2) is
valid Namely
119891 (119911) = int
1
minus1
119911 (1 + 1199112)
(1 + 1199112)2
minus 411991121199052119889] (119905) ] isin 119875
[minus11] (53)
Function 119891 has the Taylor series expansion
119891 (119911) = sum
119899 is odd119886119899119911119899 119886119899= int
1
minus1
119880119899minus1
(119905) 119889] (119905) (54)
The following inequalities are obvious
minus (119899 + 1)2le 1198672(119899) le 0 for even 119899
1198672(119899) le (119899 + 1)
2minus 1 for odd 119899
(55)
equalities hold for 119891(119911) = 119911(1 + 1199112)(1 minus 119911
2)2
For a given class 119860 sub A let us denote by Ψ119899(119860) 119899 ge 1
the set (119886119899 119886119899+2
) 119891 isin 119860 From (53) it follows that Ψ119899(119879) is
the closed convex hull of the curve
120582 [minus1 1] ni 119905 997888rarr (119880119899minus1
(119905) 119880119899+1
(119905)) (56)
FromTheorem 13 and from the equivalence
(119886119899 119886119899+2
) isin Ψ119899(119879) lArrrArr (119886
119899 119886119899+2
) isin Ψ119899(119879(2)) (57)
we get
min 119886119899119886119899+2
119891 isin 119879(2) = minus1 (58)
Hence for odd 119899 we know that
min 1198672(119899) 119891 isin 119879
(2) ge minus1 (59)
The equality holds for functions (48) or (49) providing that120572119896
= 120572119899+1minus119896
Then connecting the components of theseformulae in pairs we obtain
120572119896
119911
1 minus 2119911 cos (119896120587 (119899 + 1)) + 1199112
+ 120572119899+1minus119896
119911
1 minus 2119911 cos ((119899 + 1 minus 119896) 120587 (119899 + 1)) + 1199112
= 2120572119896
119911 (1 + 1199112)
(1 + 1199112)2
minus 41199112cos2 (119896120587 (119899 + 1))
(60)
With help of the argument given in the proof of Theorem 13we eventually obtain the odd functions for which 119886
119899119886119899+2
=
minus1
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Abstract and Applied Analysis 7
References
[1] C Pommerenke ldquoOn the coefficients and Hankel determinantsof univalent functionsrdquo Journal of the London MathematicalSociety vol 41 pp 111ndash122 1966
[2] C Pommerenke ldquoOn the Hankel determinants of univalentfunctionsrdquoMathematika vol 14 pp 108ndash112 1967
[3] A Janteng S A Halim and M Darus ldquoCoefficient inequalityfor a function whose derivative has a positive real partrdquo Journalof Inequalities in Pure and Applied Mathematics vol 7 no 2article 50 2006
[4] A Janteng S A Halim andMDarus ldquoHankel determinant forstarlike and convex functionsrdquo International Journal of Mathe-matical Analysis vol 1 no 13 pp 619ndash625 2007
[5] S K Lee V Ravichandran and S Supramaniam ldquoBounds forthe second Hankel determinant of certain univalent functionsrdquoJournal of Inequalities and Applications vol 2013 article 2812013
[6] D Vamshee Krishna andT Ramreddy ldquoHankel determinant forstarlike and convex functions of order alphardquo Tbilisi Mathemat-ical Journal vol 5 no 1 pp 65ndash76 2012
[7] C Selvaraj and T R K Kumar ldquoSecond Hankel determinantfor certain classes of analytic functionsrdquo International Journalof Applied Mathematics vol 28 no 1 pp 37ndash50 2015
[8] W Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis Conference Proceedings and LectureNotes in Analysis Vol 1 pp 157ndash169 International PressTianjin China 1992
[9] J Sokoł and J Stankiewicz ldquoRadius of convexity of some sub-classes of strongly starlike functionsrdquoZeszytyNaukowe Politech-niki Rzeszowskiej Matematyka vol 19 pp 101ndash105 1996
[10] M-S Liu J-F Xu and M Yang ldquoUpper bound of second Han-kel determinant for certain subclasses of analytic functionsrdquoAbstract and Applied Analysis vol 2014 Article ID 603180 10pages 2014
[11] A K Mishra and P Gochhayat ldquoSecond Hankel determinantfor a class of analytic functions defined by fractional deriva-tiverdquo International Journal of Mathematics and MathematicalSciences vol 2008 Article ID 153280 10 pages 2008
[12] J W Noonan and D K Thomas ldquoOn the Hankel determinantsof areally mean 119901-valent functionsrdquo Proceedings of the LondonMathematical Society vol 25 pp 503ndash524 1972
[13] G Singh ldquoHankel determinant for a new subclass of analyticfunctionsrdquo Scientia Magna vol 8 no 4 pp 61ndash65 2012
[14] G Singh ldquoHankel determinant for analytic functions withrespect to other pointsrdquo Engineering Mathematics Letters vol2 no 2 pp 115ndash123 2013
[15] T Hayami and S Owa ldquoGeneralized hankel determinant forcertain classesrdquo International Journal of Mathematical Analysisvol 4 no 52 pp 2573ndash2585 2010
[16] K O Babalola ldquoOnH3(1) Hankel determinants for some classes
of univalent functionsrdquo in Inequality Theory and Applicationsvol 6 17 Nova Science Publishers New York NY USA 2010
[17] G Shanmugam B Adolf Stephen and K O Babalola ldquoThirdHankel determinant for 120572-starlike functionsrdquo Gulf Journal ofMathematics vol 2 no 2 pp 107ndash113 2014
[18] P T Mocanu ldquoUne proprit de convexit gnralise dans la thoriede la representation conformerdquoMathematica (Cluj) vol 11 no34 pp 127ndash133 1969
[19] M S Robertson ldquoOn the coefficients of a typically-real func-tionrdquo Bulletin of the American Mathematical Society vol 41 no8 pp 565ndash572 1935
[20] D J Hallenbeck and T H MacGregor Linear Problems andConvexity Techniques in Geometric Function Theory Mono-graphs and Studies in Mathematics 22 Pitman AdavancedPublishing Program Boston Mass USA 1984
[21] W Ma ldquoGeneralized Zalcman conjecture for starlike and typ-ically real functionsrdquo Journal of Mathematical Analysis andApplications vol 234 no 1 pp 328ndash339 1999
[22] W Szapiel Extremal problems for convex sets Applications toholomorphic functions [Dissertation XXXVII] UMCS PressLublin Poland reviewer J Krzyz 1986 (Polish)
[23] W Szapiel ldquoExtreme points of convex sets (II) Inuence of nor-malization on integral representationrdquo Bulletin de lrsquoAcademiePolonaise des Sciences Serie des SciencesMathematiques vol 29pp 535ndash544 1981
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
Abstract and Applied Analysis 7
References
[1] C Pommerenke ldquoOn the coefficients and Hankel determinantsof univalent functionsrdquo Journal of the London MathematicalSociety vol 41 pp 111ndash122 1966
[2] C Pommerenke ldquoOn the Hankel determinants of univalentfunctionsrdquoMathematika vol 14 pp 108ndash112 1967
[3] A Janteng S A Halim and M Darus ldquoCoefficient inequalityfor a function whose derivative has a positive real partrdquo Journalof Inequalities in Pure and Applied Mathematics vol 7 no 2article 50 2006
[4] A Janteng S A Halim andMDarus ldquoHankel determinant forstarlike and convex functionsrdquo International Journal of Mathe-matical Analysis vol 1 no 13 pp 619ndash625 2007
[5] S K Lee V Ravichandran and S Supramaniam ldquoBounds forthe second Hankel determinant of certain univalent functionsrdquoJournal of Inequalities and Applications vol 2013 article 2812013
[6] D Vamshee Krishna andT Ramreddy ldquoHankel determinant forstarlike and convex functions of order alphardquo Tbilisi Mathemat-ical Journal vol 5 no 1 pp 65ndash76 2012
[7] C Selvaraj and T R K Kumar ldquoSecond Hankel determinantfor certain classes of analytic functionsrdquo International Journalof Applied Mathematics vol 28 no 1 pp 37ndash50 2015
[8] W Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis Conference Proceedings and LectureNotes in Analysis Vol 1 pp 157ndash169 International PressTianjin China 1992
[9] J Sokoł and J Stankiewicz ldquoRadius of convexity of some sub-classes of strongly starlike functionsrdquoZeszytyNaukowe Politech-niki Rzeszowskiej Matematyka vol 19 pp 101ndash105 1996
[10] M-S Liu J-F Xu and M Yang ldquoUpper bound of second Han-kel determinant for certain subclasses of analytic functionsrdquoAbstract and Applied Analysis vol 2014 Article ID 603180 10pages 2014
[11] A K Mishra and P Gochhayat ldquoSecond Hankel determinantfor a class of analytic functions defined by fractional deriva-tiverdquo International Journal of Mathematics and MathematicalSciences vol 2008 Article ID 153280 10 pages 2008
[12] J W Noonan and D K Thomas ldquoOn the Hankel determinantsof areally mean 119901-valent functionsrdquo Proceedings of the LondonMathematical Society vol 25 pp 503ndash524 1972
[13] G Singh ldquoHankel determinant for a new subclass of analyticfunctionsrdquo Scientia Magna vol 8 no 4 pp 61ndash65 2012
[14] G Singh ldquoHankel determinant for analytic functions withrespect to other pointsrdquo Engineering Mathematics Letters vol2 no 2 pp 115ndash123 2013
[15] T Hayami and S Owa ldquoGeneralized hankel determinant forcertain classesrdquo International Journal of Mathematical Analysisvol 4 no 52 pp 2573ndash2585 2010
[16] K O Babalola ldquoOnH3(1) Hankel determinants for some classes
of univalent functionsrdquo in Inequality Theory and Applicationsvol 6 17 Nova Science Publishers New York NY USA 2010
[17] G Shanmugam B Adolf Stephen and K O Babalola ldquoThirdHankel determinant for 120572-starlike functionsrdquo Gulf Journal ofMathematics vol 2 no 2 pp 107ndash113 2014
[18] P T Mocanu ldquoUne proprit de convexit gnralise dans la thoriede la representation conformerdquoMathematica (Cluj) vol 11 no34 pp 127ndash133 1969
[19] M S Robertson ldquoOn the coefficients of a typically-real func-tionrdquo Bulletin of the American Mathematical Society vol 41 no8 pp 565ndash572 1935
[20] D J Hallenbeck and T H MacGregor Linear Problems andConvexity Techniques in Geometric Function Theory Mono-graphs and Studies in Mathematics 22 Pitman AdavancedPublishing Program Boston Mass USA 1984
[21] W Ma ldquoGeneralized Zalcman conjecture for starlike and typ-ically real functionsrdquo Journal of Mathematical Analysis andApplications vol 234 no 1 pp 328ndash339 1999
[22] W Szapiel Extremal problems for convex sets Applications toholomorphic functions [Dissertation XXXVII] UMCS PressLublin Poland reviewer J Krzyz 1986 (Polish)
[23] W Szapiel ldquoExtreme points of convex sets (II) Inuence of nor-malization on integral representationrdquo Bulletin de lrsquoAcademiePolonaise des Sciences Serie des SciencesMathematiques vol 29pp 535ndash544 1981
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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