Abstract

We establish some new Hermite-Hadamard type integral inequalities for functions whose second-order mixed derivatives are coordinated --convex. An expression form of Hermite-Hadamard type integral inequalities via the beta function and the hypergeometric function is also presented. Our results provide a significant complement to the work of Wu et al. involving the Hermite-Hadamard type inequalities for coordinated --convex functions in an earlier article.

1. Introduction

Let be a convex mapping. Then for any with , we have the following double inequality:

This celebrated inequality is known in the literature as the Hermite-Hadamard inequality. As we all know, some of the classical inequalities for means can be derived from (1) for appropriate particular selections of the mapping . Indeed, Hermite-Hadamard’s inequality (1) has already found many applications in mathematical analysis and optimization (see, for example, [1–9]).

In recent years, the applications of various properties of extended convex functions in establishing and improving Hermite-Hadamard type inequalities have attracted the attention of many researchers (see [10–15] and references cited therein).

In [16], Wu et al. established some Hermite-Hadamard type inequalities under the assumption that the function is a coordinated --convex function. Motivated by the ideas of work [16], in this paper we study Hermite-Hadamard type inequalities related to the convexity of second-order mixed derivatives of . More precisely, we focus on establishing some new Hermite-Hadamard type inequalities for functions whose second-order mixed derivatives are coordinated --convex. For convenience of our discussions in subsequent sections, we begin with recalling some relevant definitions.

Definition 1. A function is said to be convex function if holds for all and

Definition 2 (see [5]). We say that a map belongs to the class if it is nonnegative and for all and satisfies the following inequality: In [17], the concept of -convex functions was introduced as follows.

Definition 3 (see [17]). For and , if is valid for all and , then we say that is a -convex function on .
In [18], the concept of -convex functions was presented as follows.

Definition 4 (see [18]). Let . A function is said to be -convex (in the second sense) if holds for all and

Definition 5 (see [19]). For , a function is said to be -convex if holds for all and

Definition 6 (see [20]). For some , a function is said to be extended -convex if is valid for all and .
Dragomir [21] and Dragomir and Pearce [22] considered the convexity of a function on the coordinates and put forward the following definition.

Definition 7 (see [21, 22]). A function is said to be convex on the coordinates on with and if the partial functions are convex for all and .
It should be noted that a formal definition for coordinated convex functions is stated as follows.

Definition 8 (see [21, 22]). A function is said to be convex on the coordinates on with and if the partial function holds for all .

Definition 9 (see [16]). For some and , a function is said to be coordinated --convex on with and , if holds for all , and .
Dragomir [21] and Dragomir and Pearce [22] established the following result.

Theorem 10 (see [21, 22]). Let be convex on the coordinates on with and . Then, one has the inequalities:

In this paper, we shall establish some new integral inequalities of Hermite-Hadamard type for coordinated --convex functions.

2. Lemma

Lemma 11 (see [23]). If has partial derivatives and with and , then where

3. Main Results

In this section, we establish some Hermite-Hadamard type integral inequalities for functions whose second-order mixed derivatives are coordinated --convex on the plane .

Theorem 12. Suppose that the function has continuous partial derivatives of the second-order and with , , for some and . If is coordinated --convex functions on for , then(1)if , we have (2)if , we havewhere

Proof. By Lemma 11 and Hölder’s integral inequality, we have A straightforward computation gives Now, by using the coordinated --convexity of , it follows that if , we have and if , we have By a similar argument, we obtain Applying (18) and inequalities (19)–(21) into inequality (17), we get (14) and (15). This completes the proof of Theorem 12.

Corollary 13. Under the assumptions of Theorem 12, if , then (1)if , then (2)if , then