Some New Inequalities of Hermite-Hadamard Type for (𝛼₁,𝑚₁)-(𝛼₂,𝑚₂)-Convex Functions on Coordinates

Abstract

The author introduces a new concept “()-()-convex functions on coordinates” and establishes some new inequalities of Hermite-Hadamard type for ()-()-convex functions of two variables on co-ordinates.

1. Introduction

The following definitions are well known in the literature.

Definition 1. Let be a function on an interval . If is valid for all and , then we say that is a convex function on .

Definition 2 (see [1]). For and , if is valid for all and , then we say that is an -convex function on .

Definition 3 (see [2]). For and , if is valid for all and , then we say that is an -convex function on .

Definition 4 (see [3, 4]). For and , a function is said to be convex on coordinates, if the partial mappings are convex for all and .

A formal definition for coordinated convex functions may be stated as follows.

Definition 5. A function is said to be convex on coordinates, if holds for all , .

Some inequalities of Hermite-Hadamard type for convex functions on coordinates may be recited as follows.

Theorem 6 (see Theorem 1 in [3, 4]). Let be a convex function on coordinates. Then

Theorem 7 (see [5, Theorem 2.3]). Let be a partial differentiable function. If is a convex function on coordinates, then where

Theorem 8 (see [6, Theorem 2]). Let be a partial differentiable function. If is a convex function on coordinates, then where

For more information on inequalities of Hermite-Hadamard type for various --convex functions on coordinates and for extended -convex functions on coordinates, please refer to the recently published articles [7, 8] and related references therein.

The aim of this paper is to introduce a new concept “--convex functions on coordinates” and to establish some new inequalities of Hermite-Hadamard type for --convex functions of two variables on co-ordinates.

2. A Definition and a Lemma

We now introduce a new notion “--convex functions on coordinates.”

Definition 9. For some , a function is said to be --convex on coordinates, if holds for all , .

In order to establish some new inequalities of Hermite-Hadamard type for --convex functions of two variables on coordinates, we need the following lemma.

Lemma 10. Let be a partial differentiable function and denote the set of all Lebesgue integrable functions on . If , then

Proof. Integrating by part yields Choosing and for and multiplying by on both sides of the above equations lead to the identity (12). The proof of Lemma 10 is complete.

3. Some Integral Inequalities of Hermite-Hadamard Type

We are now in a position to establish new inequalities of Hermite-Hadamard type for --convex functions on coordinates on a rectangle in the plane .

Theorem 11. Let be a partial differentiable function and . If is --convex on coordinates on with and for some and , then

Proof. Using Lemma 10, the coordinated --convexity of , and Hölder integral inequality yields Theorem 11 is thus proved.

Corollary 12. Under conditions of Theorem 11, when , one has

Corollary 13. Under conditions of Theorem 11, (1)if and , then (2)if , then

Theorem 14. Let be a partial differentiable function and . If is --convex on coordinates on with and for and , then

Proof. By Lemma 10, the coordinated --convexity of , and Hölder integral inequality, we have Theorem 14 is thus proved.

Corollary 15. Under conditions of Theorem 14, if and , then if , then

Theorem 16. Let be a partial differentiable function and . If is --convex on co-ordinates on with and for , , and , , then

Proof. Using Lemma 10, the coordinated --convexity of , and Hölder integral inequality reveals Hence, the proof of Theorem 16 is completed.

Corollary 17. Under conditions of Theorem 16, (1)if , then (2)if , , and , one has (3)if and , one has

Corollary 18. Under conditions of Theorem 16, if , then

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author appreciates the anonymous referee for his/her careful corrections to and valuable comments on the original version of this paper. This work was partially supported by the NNSF under Grant no. 11361038 of China, by the Foundation of the Research Program under Grant no. NJZY13159 of Science and Technology at Universities of Inner Mongolia Autonomous Region, and by the Science Research Funding under Grant no. NMD1225 of the Inner Mongolia University for Nationalities, China.