Petrović’s inequality on coordinates and related results · Petrović’s inequality on coordinates and related results Atiq Ur Rehman 1*, Muhammad Mudessir , Hafiza Tahira Fazal2
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Rehman et al., Cogent Mathematics (2016), 3: 1227298http://dx.doi.org/10.1080/23311835.2016.1227298
PURE MATHEMATICS | RESEARCH ARTICLE
Petrović’s inequality on coordinates and related resultsAtiq Ur Rehman1*, Muhammad Mudessir1, Hafiza Tahira Fazal2 and Ghulam Farid1
Abstract: In this paper, the authors extend Petrović’s inequality to coordinates in the plane. The authors consider functionals due to Petrović’s inequality in plane and discuss its properties for certain class of coordinated log-convex functions. Also, the authors proved related mean value theorems.
1. IntroductionA function f : [a, b] → ℝ is called mid-convex or convex in Jensen sense if for all x, y ∈ [a, b], the inequality
is valid.
In 1905, J. Jensen was the first to define convex functions using above inequality (see, Jensen, 1905; Robert & Varberg, 1974, p. 8) and draw attention to their importance.
f(x + y
2
)≤f (x) + f (y)
2
*Corresponding author: Atiq Ur Rehman, Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan E-mail: [email protected]
Reviewing editor:Lishan Liu, Qufu Normal University, China
Additional information is available at the end of the article
ABOUT THE AUTHORSAtiq Ur Rehman and Ghulam Farid are assistant professors in the Department of Mathematics at the COMSATS Institute of Information Technology (CIIT), Attock, Pakistan. Their primary research interests include real functions, mathematical inequalities, and difference equation.
Muhammad Mudessir has successfully completed his MS degree in mathematics from CIIT in this year. He is a teacher in Government Pilot Secondary School, Attock, Pakistan. His area of research includes convex analysis and inequalities in mathematics.
Hafiza Tahira Fazal received her master of philosophy degree from National College of Business Administration and Economics, Lahore, Pakistan. She is working as a lecturer in the Department of Mathematics at the University of Lahore, Sargodha, Pakistan from last two years. Her area of research includes inequalities in mathematics.
PUBLIC INTEREST STATEMENTA real-valued function defined on an interval is called convex if the line segment between any two points on the graph of the function lies above or on the graph. Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. One of the important subclass of convex functions is log-convex functions. Apparently, it would seem that log-convex functions would be unremarkable because they are simply related to convex functions. But they have some surprising properties. Recently, the concept of convex functions has been generalized by many mathematicians and different functions related or close to convex functions are defined. In this work, the variant of Petrovic’s inequality for convex functions on coordinates is given. Few generalization of the results related to it are given.
Received: 19 July 2016Accepted: 17 August 2016Published: 13 September 2016
Rehman et al., Cogent Mathematics (2016), 3: 1227298http://dx.doi.org/10.1080/23311835.2016.1227298
Definition 1 A function f : [a, b] → ℝ is said to be convex if
holds, for all x, y ∈ [a, b] and t ∈ [0, 1]. A function f is said to be strictly convex if strict inequality holds in (1.1).
A mapping f : Δ → ℝ is said to be convex in Δ if
for all (x, y), (z,w) ∈ Δ, where Δ: = [a, b] × [c,d] ⊂ ℝ2 and t ∈ [0,1].
In Dragomir (2001) gave the definition of convex functions on coordinates as follows.
Definition 2 Let Δ = [a, b] × [c,d] ⊆ ℝ2 and f : Δ → ℝ be a mapping. Define partial mappings
and
Then f is said to be convex on coordinates (or coordinated convex) in Δ if fy and fx are convex on [a, b] and [c, d] respectively for all x ∈ [a, b] and y ∈ [c,d]. A mapping f is said to be strictly convex on coordinates (or strictly coordinated convex) in Δ if fy and fx are strictly convex on [a, b] and [c, d] respectively for all x ∈ [a, b] and y ∈ [c,d].
One of the important subclass of convex functions is log-convex functions. Apparently, it would seem that log-convex functions would be unremarkable because they are simply related to convex functions. But they have some surprising properties. The Laplace transform of a non-negative func-tion is a log-convex. The product of log-convex functions is log-convex. Due to their interesting prop-erties, the log-convex functions appear frequently in many problems of classical analysis and probability theory, e.g. see (Farid, Marwan, & Rehman, 2015; Niculescu, 2012; Noor, Qi, & Awan, 2013; Pečarić, & Rehman, 2008a, 2008b; Xi & Qi, 2015; Zhang & Jiang, 2012) and the references therein.
Definition 3 A function f : I → ℝ+ is called log-convex on I if
where 𝛼, 𝛽 > 0 with � + � = 1 and x, y ∈ I.
Definition 4 (Alomari & Darus, 2009) A function f : Δ → ℝ+ is called log-convex on coordinates in Δ
if partial mappings defined in (1.2) and (1.3) are log-convex on [a, b] and [c, d], respectively, for all x ∈ [a, b] and y ∈ [c,d].
Remark 1 Every log-convex function is log convex on coordinates but the converse is not true in gen-eral. For example, f : [0, 1]2 → [0,∞) defined by f (x, y) = exy is log-convex on coodinates but not log-convex.
In Pečarić, Proschan, and Tong (1992, p. 154), Petrović’s inequality for convex function is stated as follows.
Theorem 1 Let [0,a) ⊂ ℝ, (x1,… , xn) ∈ (0, a]n and (p1,… , pn) be nonnegative n-tuples such that
(1.1)f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y)
f (tx + (1 − t)z, ty + (1 − t)w) ≤ tf (x, y) + (1 − t)f (z,w)
(1.2)fy : [a, b] → ℝ by fy(u) = f (u, y)
(1.3)fx: [c,d] → ℝ by fx(v) = f (x, v).
f (�x + �y) ≤ f �(x)f � (y)
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If f is a convex function on [0, a), then the inequality
is valid.
Remark 2 If f is strictly convex, then strict inequality holds in (1.4) unless x1 = ⋯ = xn and ∑n
i=1 pi = 1.
Remark 3 For pi = 1 (i = 1,… ,n), the above inequality becomes
This was proved by Petrović in 1932 (see Petrović, 1932).
In this paper, we extend Petrović’s inequality to coordinates in the plane. We consider functionals due to Petrović’s inequality in plane and discuss its properties for certain class of coordinated log-convex functions. Also we proved related mean value theorems.
2. Main resultsIn the following theorem, we give our first result that is Petrović’s inequality for coordinated convex functions.
Theorem 2 Let Δ = [0, a) × [0, b) ⊂ ℝ2, (x1,… , xn) ∈ (0, a]n, (y1,… , yn) ∈ (0, b]n, (p1,… , pn) and
(q1,… , qn) be non-negative n-tuples such that ∑n
i=1 pixi ≥ xj and ∑n
i=1 qiyi ≥ yj for j = 1,… ,n. Also let ∑n
i=1 pixi ∈ [0, a),∑n
i=1 pi ≥ 1 and ∑n
i=1 qiyi ∈ [0, b). If f : Δ → ℝ is coordinated convex function, then
Proof Let fx: [0, b) → ℝ and fy : [0,a) → ℝ be mappings such that fx(v) = f (x, v) and fy(u) = f (u, y). Since f is coordinated convex on Δ, therefore fy is convex on [0, a). By Theorem 1, one has
By setting y = yj, we have
this gives
Again, using Theorem 1 on terms of right-hand side for second coordinates, we have
n∑i=1
pixi ⩾ xj for j = 1, 2, 3,… ,n and
n∑i=1
pixi ∈ [0, a).
(1.4)n∑i=1
pif (xi) ⩽ f
(n∑i=1
pixi
)+
(n∑i=1
pi − 1
)f (0)
(1.5)n∑i=1
f (xi) ⩽ f
(n∑i=1
xi
)+ (n − 1)f (0).
(2.1)
n∑i,j=1
piqjf (xi , yj) ⩽ f
(n∑i=1
pixi ,
n∑j=1
qjyj
)+
(n∑j=1
qj − 1
)f
(n∑i=1
pixi , 0
)
+
(n∑i=1
pi − 1
)[f
(0,
n∑j=1
qjyj
)+
(n∑j=1
qj − 1
)f (0, 0)
].
n∑i=1
pify(xi) ⩽ fy
(n∑i=1
pixi
)+
(n∑i=1
pi − 1
)fy(0).
n∑i=1
pif (xi , yj) ⩽ f
(n∑i=1
pixi , yj
)+
(n∑i=1
pi − 1
)f (0, yj),
(2.2)n∑i=1
n∑j=1
piqjf (xi , yj) ⩽
n∑j=1
qjf
(n∑i=1
pixi , yj
)+
(n∑i=1
pi − 1
)n∑j=1
qjf (0, yj).
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and
Using above inequities in (2.2), we get inequality (2.1). ✷
Remark 4 If f is strictly coordinated convex, then above inequality is strict unless all xi’s and yi’s are not equal or
∑n
i=1 pi ≠ 1 and ∑n
j=1 qj ≠ 1.
Remark 5 If we take yi = 0 and qj = 1, (i, j = 1,...,n) with f (xi , 0) ↦ f (xi), then we get inequality (1.4).
Let I ⊆ ℝ be an interval and f : I → ℝ be a function. Then for distinct points ui ∈ I, i = 0, 1, 2. The divided differences of first and second order are defined as follows:
The values of the divided differences are independent of the order of the points u0,u1,u2 and may be extended to include the cases when some or all points are equal, that is
provided that f ′ exists. Now passing the limit u1 → u0 and replacing u2 by u in second-order divided difference, we have
provided that f ′ exists. Also, passing to the limit ui → u (i = 0, 1, 2) in second-order divided differ-ence, we have
provided that f ′′ exists.
One can note that, if for all u0,u1 ∈ I, [u0,u1, f ] ≥ 0, then f is increasing on I and if for all u0,u1,u2 ∈ I, [u0,u1,u2, f ] ≥ 0, then f is convex on I.
Now we define some families of parametric functions which we use in sequal.
Let I = [0,a) and J = [0, b) be intervals and let for t ∈ (c,d) ⊆ ℝ, ft: I × J → ℝ be a mapping. Then we define functions
and
n∑j=1
qjf
(n∑i=1
pixi , yj
)⩽ f
(n∑i=1
pixi ,
n∑j=1
qjyj
)+
(n∑j=1
qj − 1
)f
(n∑i=1
pixi , 0
)
(n∑i=1
pi − 1
)n∑j=1
qjf (0, yj) ⩽
(n∑i=1
pi − 1
)[f
(0,
n∑j=1
qjyj
)+
(n∑j=1
qj − 1
)f (0, 0)
].
(2.3)[ui ,ui+1, f ] =
f (ui+1) − f (ui)
ui+1 − ui, (i = 0, 1),
[u0,u
1,u
2, f ] =
[u1,u
2, f ] − [u
0,u
1, f ]
u2− u
0
.
(2.5)[u0,u
0, f ] = lim
u1→u
0
[u0,u
1, f ] = f �(u
0)
(2.6)[u0,u
0,u, f ] = lim
u1→u
0
[u0,u
1,u, f ] =
f (u) − f (u0) − (u − u
0)f �(u
0)
(u − u0)2
,u ≠ u0
(2.7)[u,u,u, f ] = limui→u
[u0,u
1,u
2, f ] =
f ��(u)
2
ft,y : I → ℝ by ft,y(u) = ft(u, y)
(2.4)
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where x ∈ I and y ∈ J.
Suppose denotes the class of functions ft: I × J → ℝ for t ∈ (c,d) such that
and
are log-convex functions in Jensen sense on (c, d) for all x ∈ I and y ∈ J.
We define linear functional � (f ) as a non-negative difference of inequality (2.1)
Remark 6 Under the assumptions of Theorem 2, if f is coordinated convex in Δ, then � (f ) ≥ 0.
The following lemmas are given in Pečarić and Rehman (2008b).
Lemma 1 Let I ⊆ ℝ be an interval. A function f : I → (0,∞) is log-convex in Jensen sense on I, that is, for each r, t ∈ I
if and only if the relation
holds for each m,n ∈ ℝ and r, t ∈ I.
Lemma 2 If f is convex function on interval I then for all x1, x2, x3 ∈ I for which x1 < x2 < x3, the fol-lowing inequality is valid:
Our next result comprises properties of functional defined in (2.8).
Theorem 3 Let the functional � defined in (2.8) and ft ∈ . Then the following are valid:
(a) The function t ↦ � (ft) is log-convex in Jensen sense on (c, d).
(b) If the function t ↦ � (ft) is continuous on (c, d), then it is log-convex on (c, d).
(c) If � (ft) is positive, then for some r < s < t, where r, s, t ∈ (c,d), one has
Rehman et al., Cogent Mathematics (2016), 3: 1227298http://dx.doi.org/10.1080/23311835.2016.1227298
Proof
(a) Let
where m,n ∈ ℝ and t, r ∈ (c,d). We can consider
and
Now we take
As [u0,u1,u2, ft,y] is log convex in Jensen sense, so using Lemma 1, the right-hand side of above ex-pression is non-negative, so hy is convex on I. Similarly, one can show that hx is also convex on J, which concludes h is coordinated convex on Δ. By Remark 6, � (h) ≥ 0, that is,
so t ↦ � (ft) is log-convex in Jensen sense on (c, d).
(b) Additionally, we have t ↦ � (ft) is continuous on (c, d), hence we have t ↦ � (ft) is log-convex on (c, d).
(c) Since t ↦ � (ft) is log-convex on (c, d), therefore for r, s, t ∈ (c,d) with r < s < t and f (t) = log� (t) in Lemma 2, we have
which is equivalent to (2.9). ✷
Example 1 Let t ∈ (0,∞) and �t:[0,∞)2→ ℝ be a function defined as
Define partial mappings
and
As we have
h(u, v) = m2ft(u, v) + 2mnf t+r2
(u, v) + n2fr(u, v)
hy(u) = m2ft,y(u) + 2mnf t+r
2,y(u) + n
2fr,y(u)
hx(v) = m2ft,x(v) + 2mnf t+r
2,x(v) + n
2fr,x(v).
[u0,u1,u2,hy] = m2[u0,u1,u2, ft,y] + 2mn[u0,u1,u2, f t+r
Rehman et al., Cogent Mathematics (2016), 3: 1227298http://dx.doi.org/10.1080/23311835.2016.1227298
This gives t ↦ [u0,u0,u0,�t,v] is log-convex in Jensen sense. Similarly, one can deduce that t ↦ [v0, v0, v0,�t,u] is also log-convex in Jensen sense. If we choose ft = �t in Theorem 3, we get log convexity of the functional � (�t).
In special case, if we choose �t(u, v) = �t(u, 1), then we get (Butt, Pečarić, & Rehman, A. U. 2011, Example 3).
Example 2 Let t ∈ [0,∞) and �t :[0,∞)2→ ℝ be a function defined as
Define partial mappings
and
for all u, v ∈ [0,∞).
As we have
This gives t ↦ [u0,u0,u0, �t,v] is log convex in Jensen sense. Similarly, one can deduce that t ↦ [v0, v0, v0, �t,u] is also log-convex in Jensen sense. If we choose ft = �t in Theorem 3, we get log convexity of the functional � (�t).
In special case, if we choose �t(u, v) = �t(u, 1), then we get (Butt et al., 2011, Example 8).
Example 3 Let t ∈ [0,∞) and �t :[0,∞)2→ ℝ be a function defined as
Define partial mappings
and
As we have
This gives t ↦ [u0,u0,u0, �t,v] is log-convex in Jensen sense. Similarly one can deduce that t ↦ [v0, v0, v0, �t,u] is also log-convex in Jensen sense. If we choose ft = �t in Theorem 3, we get log convexity of the functional � (�t).
(2.11)�t(u, v) =
{uveuvt
t, t ≠ 0,
u2v2, t = 0.
�t,v :[0,∞) → ℝ by �t,v(u) = �t(u, v)
�t,u:[0,∞) → ℝ by �t,u(v) = �t(u, v)
[u,u,u, �t,v] =�2�t,v
�u2= euvt(2v2 + uv2) ≥ 0 ∀ t ∈ (0,∞).
(2.12)�t(u, v) =
{euvt
t, t ≠ 0,
uv, t = 0.
�t,v :[0,∞) → ℝ by �t,v(u) = �t(u, v)
�t,u:[0,∞) → ℝ by �t,u(v) = �t(u, v).
[u,u,u, �t,v] =�2�t,v
�u2= tv2euvt ≥ 0 ∀ t ∈ (0,∞).
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In special case, if we choose �t(u, v) = �t(u, 1), then we get (Butt et al., 2011, Example 9).
3. Mean value theoremsIf a function is twice differentiable on an interval I, then it is convex on I if and only if its second order derivative is non-negative. If a function f (X): = f (x, y) has continuous second-order partial deriva-tives on interior of Δ, then it is convex on Δ if the Hessian matrix
is non-negative definite, that is, vHf (X)vt is non-negative for all real non-negative vector v.
It is easy to see that f : Δ → ℝ is coordinated convex on Δ iff
are non-negative for all interior points (x, y) in Δ2.
Lemma 3 Let f :Δ → ℝ be a function such that
and
for all interior points (x, y) in Δ2. Consider the function �1,�
2: Δ → ℝ defined as
then �1,�2 are convex on coordinates in Δ.
Proof Since
and
for all interior points (x, y) in Δ, �1 is convex on coordinates in Δ. Similarly, one can prove that �2 is also convex on coordinates in Δ. ✷
Theorem 4 Let f : Δ → ℝ be a mapping which has continuous partial derivatives of second order in Δ and �(x, y): = x2 + y2. Then, there exist (�1, �1) and (�2, �2) in the interior of Δ such that
Hf (X) =⎛⎜⎜⎝
�2f (X)
�x2�2f (X)
�y�x�2f (X)
�x�y
�2f (X)
�y2
⎞⎟⎟⎠
f ��x (y) =�2f (x, y)
�y2and f ��y (x) =
�2f (x, y)
�x2
m1 ≤�2f (x, y)
�x2≤ M1
m2 ≤�2f (x, y)
�y2≤ M2
�1 =1
2max{M1,M2}(x
2+ y2) − f (x, y)
�2 = f (x, y) −1
2min{m1,m2}(x
2+ y2)
�2�1(x, y)
�x2= max{M1,M2} −
�2f (x, y)
�x2≥ 0
�2�1(x, y)
�y2= max{M1,M2} −
�2f (x, y)
�y2≥ 0
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and
provided that � (�) is non-zero.
Proof Since f has continuous partial derivatives of second order in Δ, there exist real numbers m1,m2,M1 and M2 such that
for all (x, y) ∈ Δ.
Now consider functions �1 and �2 defined in Lemma 3. As �1 is convex on coordinates in Δ,
that is
this leads us to
On the other hand, for function �2, one has
As � (�) ≠ 0, combining inequalities (3.1) and (3.2), we get
Then there exist (�1, �1) and (�2, �2) in the interior of Δ such that
and
hence the required result follows. ✷
Theorem 5 Let �1,�
2: Δ → ℝ be mappings which have continuous partial derivatives of second or-
der in Δ. Then there exists (�1, �1) and (�2, �2) in Δ such that
� (f ) =1
2
�2f (�1, �1)
�x2� (�)
� (f ) =1
2
�2f (�2, �2)
�y2� (�)
m1 ≤�2f (x, y)
�x2≤ M1 and m2 ≤
�2f (x, y)
�y2≤ M2,
� (�1) ≥ 0,
�
(1
2max{M1,M2}�(x, y) − f (x, y)
)≥ 0,
(3.1)2� (f ) ≤ max{M1,M2}� (�).
(3.2)min{m1,m2}� (�) ≤ 2� (f ).
min{m1,m2} ≤2� (f )
� (�)≤ max{M1,M2}.
2� (f )
� (�)=
�2f (�1, �1)
�x2
2� (f )
� (�)=
�2f (�2, �2)
�y2,
(3.3)� (�1)
� (�2)=
�2�1(�1,�1)
�x2
�2�2(�1,�1)
�x2
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and
Proof We define the mapping P: Δ → ℝ such that
where k1 = � (�2) and k2 = � (�1).
Using Theorem 4 with f = P, we have
and
Since � (�) ≠ 0, we have
and
which are equivalent to required results. ✷
(3.4)� (�1)
� (�2)=
�2�1(�2,�2)
�y2
�2�2(�2,�2)
�y2
.
P = k1�1 − k2�2,
2� (P) = 0 =
{k1
�2�1
�x2− k2
�2�2
�x2
}� (�)
2� (P) = 0 =
{k1
�2�1
�y2− k2
�2�2
�y2
}� (�).
k2k1
=
�2�1(�1,�1)
�x2
�2�2(�1,�1)
�x2
k2k1
=
�2�1(�2,�2)
�y2
�2�2(�2,�2)
�y2
,
FundingThe authors received no direct funding for this research.
Information Technology, Attock, Pakistan.2 Department of Mathematics, University of Lahore, Sargodha
Campus, Pakistan.
Citation informationCite this article as: Petrović’s inequality on coordinates and related results, Atiq Ur Rehman, Muhammad Mudessir, Hafiza Tahira Fazal & Ghulam Farid, Cogent Mathematics (2016), 3: 1227298.
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