Journal of Applied Mathematics

Journal of Applied Mathematics / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 935020 | https://doi.org/10.1155/2013/935020

Feixiang Chen, "A Note on Hermite-Hadamard Inequalities for Products of Convex Functions", Journal of Applied Mathematics, vol. 2013, Article ID 935020, 5 pages, 2013. https://doi.org/10.1155/2013/935020

A Note on Hermite-Hadamard Inequalities for Products of Convex Functions

Academic Editor: Maurizio Porfiri
Received14 Sep 2013
Accepted17 Nov 2013
Published02 Dec 2013

Abstract

We obtain some new Hermite-Hadamard type inequalities for products of convex functions. We conclude that the results obtained in this work are the refinements of the present results.

1. Introduction

If is a convex function on the interval , then, for any with , we have the following double inequality: This remarkable result is well known in the literature as the Hermite-Hadamard inequality.

Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [17]).

In [8], Pachpatte established two new Hermite-Hadamard type inequalities for products of convex functions as follows.

Theorem 1. Let and be real-valued, nonnegative, and convex functions on . Then, where and .

Some new integral inequalities involving two nonnegative and integrable functions that are related to the Hermite-Hadamard type are also established by many authors. In [9], Pachpatte established some Hermite-Hadamard type inequalities involving two log-convex functions. An analogous result for -convex functions is obtained by Kirmaci et al. in [10]. In [11], Sarikaya presented some integral inequalities for two -convex functions. Some Hermite-Hadamard type inequalities for two operator convex functions are given by Bacak and Türkmen in [12]. For recent results and generalizations concerning Hermite-Hadamard type inequality for product of two functions, see [13] and the references given therein.

In this paper, we obtain new generalizations of Hermite-Hadamard type inequalities for products of convex functions. This result refines the Hermite-Hadamard type inequalities given in Theorem 1.

2. Lemma

Lemma 2. Let and be real-valued, nonnegative, and convex functions on . If , then where and .

Proof. We get the conclusions by and are nonnegative and convex functions on and Theorem 1.

3. Main Results

Theorem 3. Let and be real-valued, nonnegative, and convex functions on . Then, where , , , and

Proof. Since and are convex on and nonnegative, by (2), we get
Firstly, and are convex on and nonnegative; applying (2) on , we get
Similarly, we can show that
Multiplying (8) by and (9) by and adding the resulting inequalities, we get
Using the fact that we have which completes the proof.

Remark 4. Applying Theorem 3 for , we get the result in Theorem 1. So, we conclude that our results give an improvement of Theorem 1.

Corollary 5. With notations above, one has the following inequality: where is defined in Theorem 3.

Theorem 6. Let and be real-valued, nonnegative, and convex functions on . Then, where   and  .

Proof. Since and are convex on and nonnegative, for , we observe that
Integrating both sides of the previous inequality with respect to over , we obtain
On the other hand, we have
By a similar way, we obtain
So, which yields the desired result.

4. Conclusion

In this paper, we establish some new Hermite-Hadamard type inequalities for products of convex functions. The results obtained in this work are the refinements of the present results. An interesting topic is whether we can use the methods in this paper to refine the Hermite-Hadamard inequality for coordinated convex functions.

Acknowledgment

This work is supported by the Youth Project of Chongqing Three Gorges University of China.

References

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Copyright © 2013 Feixiang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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