Hermite-Hadamard Type Inequalities Associated with Coordinated -Convex Functions

Abstract

In this paper, we introduce the definition of coordinated -convex function and establish some Hermite-Hadamard type integral inequalities for coordinated -convex functions.

1. Introduction

Let be a convex function on . The following integral inequalityis known in the literature as Hermite-Hadamard inequality.

The investigation on extended convex functions has become a hot research topic in recent years. The applications of various properties of extended convex functions in establishing and improving Hermite-Hadamard type inequalities especially have attracted the attention of many researchers. For convenience of our discussions in subsequent sections, let us introduce some relevant definitions and earlier results below.

We first recall the notion of quasi-convex functions which generalizes the notion of convex functions as follows.

Definition 1 (see [1, 2]). A function is said to be quasi-convex (QC) on , if holds for all and .

It is obvious that any convex function is a quasi-convex function but the reverse is not true. Because there exist quasi-convex functions which are not convex (see [3]).

Ion [3] and Alomari et al. [4] established the following Hermite-Hadamard type inequalities for quasi-convex functions, respectively.

Theorem 2. Let be a differentiable function on , with If is quasi-convex function on , then the following inequalities hold:

Alomari et al. [5] presented the following Hermite-Hadamard type inequality for twice differentiable quasi-convex functions.

Theorem 3. Let : be a differentiable function on , with If is quasi-convex function on , then the following inequality holds:

In [6], Wu et al. proved the following Hermite-Hadamard type inequality for thrice differentiable quasi-convex functions.

Theorem 4. Let be a differentiable function on , with If is quasi-convex function on , then for one has

In [1], the authors introduced the class of real functions of JQC type as follows.

Definition 5 (see [1]). A function is Jensen- or J-quasi-convex (JQC) if holds for all .

In [7], the -convex function was defined as follows.

Definition 6 (see [7]). For and , if is valid for all and , then we say that is an -convex function on .

In [8] the concept of -convex functions was presented as follows.

Definition 7 (see [8]). Let . A function is said to be -convex (in the second sense) if holds for all and

In [9] the concept of extended -convex functions was introduced as follows.

Definition 8 (see [9]). For some , a function is said to be extended -convex if is valid for all and .

Dragomir [10] and Dragomir and Pearce [11] considered the convexity of a function on the coordinates and put forward the following definition.

Definition 9 (see [10, 11]). A function is said to be convex on the coordinates on with and if the partial functions are convex when defined for all , .

It should be noted that a formal definition for coordinated convex functions is stated as follows.

Definition 10. A function is said to be convex on the coordinates on with and if the following inequalityholds for all , .

In [12] the concept of coordinated -convex functions was presented as follows.

Definition 11 (see [12]). For some , a function is called coordinated -convex on with and , if the following inequalityis valid for all , , and .

Remark 12. In (12), if and , then .

In [13], the following Hermite-Hadamard type inequality for -convex functions was proved.

Theorem 13 (see [13]). Let be -convex and . If for , then

Dragomir [10] and Dragomir and Pearce [11] established the following inequality for coordinated convex functions.

Theorem 14 (see [10, 11]). Let be convex on the coordinates on with and . Then, one has the inequalities

Besides the results mentioned above, the inequalities of Hermite-Hadamard type involving harmonically -convex functions, extended -convex functions, -geometrically convex functions, and so on can be found in [14–17]. It deserves to be noticed that, in a recent paper [18], Krnić et al. use some operator techniques with interpolation of convex functions to prove the Hermite-Hadamard type inequalities, which provides a new approach to study this type of inequalities.

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

2. Definitions and Lemmas

Firstly, we introduce the definition of coordinated -convex function on the rectangle of ().

Definition 15. For some and , a function is called coordinated -convex on with and , if the following inequalityis valid for all , , and .

To establish new Hermite-Hadamard type inequalities for coordinated -convex functions, we need the following lemma.

Lemma 16. Let , , and ; thenwhere

Proof. For , or , then (1)If and , we have(2)If and , we have (3)If and , we have (4)If and , we have Lemma 16 is thus proved.

3. Some Integral Inequalities of Hermite-Hadamard Type

In this section, we establish some integral inequalities of Hermite-Hadamard type for coordinated -convex functions on the plane ().

Theorem 17. Let and , and let the function be nonnegative. Suppose that is coordinated -convex on with , . If , then

Proof. From the coordinated -convexity of , we obtainfor all .
Integrating with respect to over and putting for , we haveSince for , by for and the coordinated -convexity of , we deduce thatNow we perform the change of variable in the above integrals; we obtainFrom inequalities (25) and (27), it follows thatSimilarly, then one obtainsUsing inequalities (28) and (29), we get (23). Theorem 17 is proved.

Corollary 18. Under the conditions of Theorem 17, if , then

Theorem 19. Suppose that is a coordinated -convex on with , , and for some and . If , thenwhere is defined by (16).

Proof. Letting for , from the coordinated -convexity of , we haveFor all , by making the change of variable = for all and using the coordinated -convexity of , we deduce thatFrom (33) and Lemma 16, we obtainUsing (32) and (34), it follows thatSimilarly, we haveUtilizing inequalities (35) and (36), we obtain the inequality (31) asserted by Theorem 19.

Corollary 20. Under the conditions of Theorem 19, if , then

Corollary 21. Under the conditions of Theorems 17 to 19, if , then

Theorem 22. Suppose that is a coordinated -convex on with , , and for some , . If , thenwhere is the classical Beta function which is defined by

Proof. Letting and for all and , from the coordinated -convexity of , we havewhich is the desired inequality (39) stated in Theorem 22.

Corollary 23. Under the conditions of Theorem 22, if , then

Corollary 24. Under the conditions of Theorem 22, if , then

Disclosure

All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China under Grant no. 11361038, the Natural Science Foundation of Fujian under Grant no. 2016J01023, the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant no. NJZY16180, and the Science Research Fund of Inner Mongolia University for Nationalities under Grant no. NMDGP1713.