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.

Subjects: Foundations & Theorems; Mathematics & Statistics; Science

Keywords: Petrović’s inequality; log-convexity; convex functions on coordinates

2000 Mathematics subject classifications: Primary 26A51; Secondary 26D15

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: atiq@mathcity.org

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

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

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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

ft,x: J → ℝ by ft,x(v) = ft(x, v),

t ↦ [u0,u1,u2, ft,y] ∀ u0,u1,u2 ∈ I

t ↦ [v0, v1, v2, ft,x] ∀ v0, v1, v2 ∈ J

(2.8)

� (f ) = 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,j=1

piqjf (xi , yj).

f (r)f (t) ≥ f 2(t + r

2

)

m2f (t) + 2mnf(t + r

2

)+ n2f (r) ≥ 0

(x3 − x2)f (x1) + (x1 − x3)f (x2) + (x2 − x1)f (x3) ≥ 0.

(2.9)[� (fs)

]t−r≤[� (fr)

]t−s[� (ft)

]s−r.

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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

2,y] + n

2[u0,u1,u2, fr,y].

m2� (ft) + 2mn� (f t+r

2

) + n2� (fr) ≥ 0,

(t − s) log� (fr) + (r − t) log� (fs) + (s − r) log� (ft) ≥ 0,

(2.10)�t(u, v) =

{utvt

t(t−1), t ≠ 1,

uv(logu + log v), t = 1.

�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= ut−2vt ≥ 0 ∀ t ∈ (0,∞).

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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)

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�2�2(�1,�1)

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,

FundingThe authors received no direct funding for this research.

Author detailsAtiq Ur Rehman1

E-mail: atiq@mathcity.orgORCID ID: http://orcid.org/0000-0002-7368-0700Muhammad Mudessir1

E-mail: mudessir001@gmail.comHafiza Tahira Fazal2

E-mail: tahiramalik1230@gmail.comGhulam Farid1

E-mails: faridphdsms@hotmail.com, ghlmfarid@ciit-attock.edu.pk1 Department of Mathematics, COMSATS Institute of

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.

ReferencesAlomari, M., & Darus, M. (2009). On the Hadamard’s inequality

for log convex functions on coordinates. Journal of Inequalities and Applications, Article ID 283147. 13. doi:10.1155/2009/283147

Butt, S., Pečarić, J., & Rehman, A. U. (2011). Exponential convexity of Petrović and related functional. Journal of Inequalities and Applications, 89, 16. doi:10.1186/1029-242X-2011-89

Dragomir, S. S. (2001). On Hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwanese Journal of Mathematics, 4, 775–788.

Farid, G., & , Marwan, M., & Rehman, A. U. (2015). New mean value theorems and generalization of Hadamard inequality via coordinated m-convex functions. Journal of Inequalities and Applications, 11. Article ID 283. doi:10.1186/s13660-015-0808-z

Jensen, J. (1905). Om konvekse Funktioner og Uligheder mellem Middelvaerdier [About convex functions and inequalities between mean values]. Nyt tidsskrift for matematik, 16B, 49–69.

Niculescu, C. P. (2012). The Hermite-Hadamard inequality for log-convex functions. Nonlinear Analysis, 75, 662–669. doi:10.1016/j.na.2011.08.066.

Noor, M. A., Qi, F., & Awan, M. U. (2013). Some Hermite-Hadamard type inequalities for log-h-convex functions. Analysis, 33, 367–375. doi:10.1524/anly.2013.1223.

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Pečarić, J., Proschan, F., & Tong, Y. L. (1992). Convex functions, partial orderings and statistical applications. New York, NY: Academic Press.

Pečarić, J., & Rehman, A. U. (2008a). On logarithmic convexity for power sums and related results. Journal of Inequalities and Applications, 12. Article ID 389410. doi:10.1155/2008/389410

Pečarić, J., & Rehman, A. U. (2008b). On logarithmic convexity for power sums and related results II. Journal of Inequalities and Applications, Article ID 305623. doi:10.1155/2008/305623

Petrović, M. (1932). Sur une fontionnelle. Publications de l’Institut Mathématique, University of Belgrade, 149–146.

Roberts, A. W., & Varberg, D. E. (1974). Convex functions (Vol. 57). Academic Press.

Xi, B. Y., & Qi, F. (2015). Integral inequalities of Hermite-Hadamard type for ((α,m),log)-convex functions on co-ordinates. Probl. Anal. Issues Anal., 4, 73–92. doi:10.15393/j3.art.2015.2829.

Zhang, X., & Jiang, W. (2012). Some properties of log-convex function and applications for the exponential function. Computers & Mathematics with Applications, 63, 1111–1116. doi:10.1016/j.camwa.2011.12.019.