Research Article | Open Access

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

**Academic Editor:**Chuanzhi Bai

#### 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
*