On Some Integral Inequalities Related to Hermite-Hadamard-Fejér Inequalities for Coordinated Convex Functions

Abstract

Several new mappings associated with coordinated convexity are proposed, by which we obtain some new Hermite-Hadamard-Fejér type inequalities for coordinated convex functions. We conclude that the results obtained in this work are the generalizations of the earlier results.

1. Introduction

Let be a convex function and with ; then is known as the Hermite-Hadamard inequality.

In [1], Fejér established the following weighted generalization of inequality (1).

Theorem 1. If is a convex function, then the inequality holds, where is positive, integrable, and symmetric about .

Inequalities (1) and (2) have been extended, generalized, and improved by a number of authors (e.g., [2–9]).

In [4], Dragomir proposed the following Hermite-Hadamard type inequalities which refine the first inequality of (1).

Theorem 2 (see [4]). Let be convex on . Then is convex, increasing on , and, for all , where

An analogous result for convex functions which refines the second inequality of (1) is obtained by Yang and Hong in [10] as follows.

Theorem 3 (see [10]). Let be convex on . Then is convex, increasing on , and, for all , where

A modification for convex functions which is also known as coordinated convex functions was introduced as follows by Dragomir in [5].

Let us consider the bidimensional interval in with and ; a mapping is said to be convex on if the inequality holds for all , and .

A function is said to be coordinated convex on if the partial mappings , , and , are convex for all and .

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

Definition 4. A function is said to be convex on coordinates on if the inequality holds for all , , , and and .

Dragomir in [5] established the following Hermite-Hadamard type inequalities for coordinated convex functions in a rectangle from the plane .

Theorem 5. Suppose that is convex on the coordinates on . Then one has the following inequalities:

The mapping connected with the first inequality of (9) is considered in [5].

If is a coordinated convex function, then the following mapping on can be defined by The mapping has the following properties:(i)is coordinated convex and monotonic nondecreasing on ;(ii)we have the following bounds for :

Recently, Hwang et al. [11] established a monotonic nondecreasing mapping connected with the Hadamard’s inequality for coordinated convex functions in a rectangle from the plane as follows.

Theorem 6 (see [11]). Suppose that is coordinated convex on and the mapping is defined by Then(i)the mapping is coordinated convex on ;(ii)the mapping is coordinated monotonic nondecreasing on ;(iii)we have the bounds

Fejér-type inequality for coordinated convex mappings is established in [12] as follows.

Theorem 7 (see [12]). Suppose that is a coordinated convex function on . Then one has the following inequalities: where is positive, integrable, and symmetric about and .

Theorem 8 (see [12]). Let be a coordinated convex function, and consider the function which is positive, integrable, and symmetric about and . Let be a function defined on by Then is a coordinated convex function on , nondecreasing on . Moreover,

Theorem 9 (see [12]). Let be a coordinated convex function, and consider the function which is positive, integrable, and symmetric about and . Let be a function defined on by Then, is coordinated convex on , symmetric about , and non-decreasing on , and

In this paper, we establish some new results about the Hermite-Hadamard-Fejér type inequality for coordinated convex mappings which generalize the results (10), (12), (15), and (17).

2. Main Results

We will use the following lemma to prove our results.

Lemma 10 (see [13]). Let be a convex function, , , , , and let be defined by , . Then is convex, increasing on , and, for all ,

Lemma 11. Let be a coordinated convex function, , , , , , , , and let be defined by , . Then is coordinated convex, coordinated monotonic nondecreasing on , and, for all ,

Proof. We note that if is convex and is linear, then the composition is convex. Also we note that a positive constant multiple of a convex function and a sum of two convex functions are convex. Hence, it is easily observed that is coordinated convex, coordinated monotonic nondecreasing on , from the coordinated convexity of and Lemma 10. Next, Also, From the coordinated convexity of , we obtain We note that , , , , , , , . So which completes the proof.

Theorem 12. Let , , , , , , , , and be defined as in Lemma 11, and let be nonnegative and integrable and Then

Proof. For every , we have the identity Since is nonnegative, multiplying (21) by , integrating the resulting inequalities over , and using (26), we have Thus, inequalities (27) follow by using the identity (28).

Remark 13. If we choose , , in Theorem 12, then inequalities (27) reduce to inequalities (14).

Remark 14. If we choose , , , and in Theorem 12, then inequalities (27) reduce to inequalities (9).

Theorem 15. Let , , , , and be defined as in Lemma 11, , , , , , , and let be defined on by Then, is coordinated convex and coordinated monotonic nondecreasing on , and

Proof. That is coordinated convex follows immediately from the coordinated convexity of . Next, the conditions and imply that and , respectively. It follows from Lemma 11 that   is coordinated monotonic nondecreasing on and hence is coordinated monotonic nondecreasing on . Finally, the last inequality of (31) follows from (21), and the proof is completed.

Similarly, we have the following theorem.

Theorem 16. Let , , , , and be defined as in Lemma 11,   , , , , , , and let be defined on by Then, is coordinated convex and coordinated monotonic nondecreasing on , and

Proof. The proof is similar to that of Theorem 15, so we omit the details.

Remark 17. Identity (10) is a special case of (30) if we choose , , .

Remark 18. Identity (12) is a special case of (32) if we choose , , .

Theorem 19. Let , , , , and be defined as in Lemma 11, , , , , , , and let be defined as in Theorem 12. Let be defined on by for some , . Then is coordinated convex and coordinated monotonic nondecreasing on , and

Proof. Since is coordinated convex and is nonnegative, we see that is coordinated convex on . Next, for each , where , , it follows from Lemma 11 that   is coordinated monotonic nondecreasing on . Using (26) we see that is coordinated monotonic nondecreasing on , which completes the proof.

Theorem 20. Let , , , , and be defined as in Lemma 11, , , , , , , and let be defined as in Theorem 12. Let be defined on by for some , . Then is coordinated convex and coordinated monotonic nondecreasing on , and

Proof. That is coordinated convex follows immediately from the coordinated convexity of . Next, for each , where , , it follows from Lemma 11 that , is coordinated monotonic nondecreasing on . Since and are increasing on , respectively, we get that is coordinated monotonic nondecreasing on . Since is nonnegative and satisfies (26), it follows that is coordinated monotonic nondecreasing on . Finally, the last inequality of (37) follows from (21), and the proof is completed.

Remark 21. Identity (15) is a special case of (34) if we choose , , .

Remark 22. Identity (17) is a special case of (36) if we choose , , .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by Youth Project of Chongqing Three Gorges University of China (no. 13QN11).