Journal of Applied Mathematics

Volume 2013 (2013), Article ID 935020, 5 pages

http://dx.doi.org/10.1155/2013/935020

Research Article

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

Feixiang Chen

School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China

Received 14 September 2013; Accepted 17 November 2013

Academic Editor: Maurizio Porfiri

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.

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., [1–7]).

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

S. Abramovich, G. Farid, and J. Pečarić, “More about Hermite-Hadamard inequalities, Cauchy's means, and superquadracity,” Journal of Inequalities and Applications, vol. 2010, Article ID 102467, 14 pages, 2010.

A. Barani, S. Barani, and S. S. Dragomir, “Refinements of Hermite-Hadamard inequalities for functions when a power of the absolute value of the second derivative is P-convex,” Journal of Applied Mathematics, vol. 2012, Article ID 615737, 10 pages, 2012.

M. Bessenyei and Z. Páles, “Hadamard-type inequalities for generalized convex functions,” Mathematical Inequalities & Applications, vol. 6, no. 3, pp. 379–392, 2003.

S. S. Dragomir, “Hermite-Hadamard's type inequalities for operator convex functions,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 766–772, 2011.

S. S. Dragomir, “Hermite-Hadamard's type inequalities for convex functions of selfadjoint operators in Hilbert spaces,” Linear Algebra and Its Applications, vol. 436, no. 5, pp. 1503–1515, 2012.

A. El Farissi, “Simple proof and refinement of Hermite-Hadamard inequality,” Journal of Mathematical Inequalities, vol. 4, no. 3, pp. 365–369, 2010.

X. Gao, “A note on the Hermite-Hadamard inequality,” Journal of Mathematical Inequalities, vol. 4, no. 4, pp. 587–591, 2010.

B. G. Pachpatte, “On some inequalities for convex functions,” RGMIA Research Report Collection, vol. 6, 2003.

B. G. Pachpatte, “A note on integral inequalities involving two log-convex functions,” Mathematical Inequalities & Applications, vol. 7, no. 4, pp. 511–515, 2004.

U. S. Kirmaci, M. Klaričić Bakula, M. E. Özdemir, and J. Pečarić, “Hadamard-type inequalities for s-convex functions,” Applied Mathematics and Computation, vol. 193, no. 1, pp. 26–35, 2007.

M. Z. Sarikaya, A. Saglam, and H. Yildirim, “On some Hadamard-type inequalities for h-convex functions,” Journal of Mathematical Inequalities, vol. 2, no. 3, pp. 335–341, 2008.

V. Bacak and R. Türkmen, “New inequalities for operator convex functions,” Journal of Inequalities and Applications, vol. 2013, article 190, 2013.

E. Set, M. E. Özdemir, and S. S. Dragomir, “On the Hermite-Hadamard inequality and other integral inequalities involving two functions,” Journal of Inequalities and Applications, vol. 2010, Article ID 148102, 9 pages, 2010.