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Research Article | Open Access

Volume 2014 |Article ID 796132 | https://doi.org/10.1155/2014/796132

Ruiyin Xiang, Feixiang Chen, "On Some Integral Inequalities Related to Hermite-Hadamard-Fejér Inequalities for Coordinated Convex Functions", Chinese Journal of Mathematics, vol. 2014, Article ID 796132, 10 pages, 2014. https://doi.org/10.1155/2014/796132

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

Accepted14 Oct 2014
Published17 Nov 2014

#### 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 , 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., ).

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

Theorem 2 (see ). 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  as follows.

Theorem 3 (see ). 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 .

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  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 .

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.  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 ). 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  as follows.

Theorem 7 (see ). 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 ). 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 ). 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 ). 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 byThen, 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 byThen, 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