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Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 921828, 6 pages

http://dx.doi.org/10.1155/2013/921828

## On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Several Functions

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121, Thailand

Received 18 March 2013; Accepted 14 May 2013

Academic Editor: Nelson Merentes

Copyright © 2013 Banyat Sroysang. 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 present some new Hermite-Hadamard-type inequalities and other integral inequalities involving several functions.

#### 1. Introduction and Preliminaries

In 2003, Pachpatte [1] gave some Hermite-Hadamard-type inequalities involving two convex functions, and then Pachpatte [2] also gave, in 2004, some Hermite-Hadamard-type inequalities involving two log-convex functions. In 2007, Kirmaci et al. [3] gave some Hadamard-type inequalities involving -convex functions. In 2008, Bakula et al. [4] presented some Hadamard-type inequalities involving -convex functions and -convex functions. In 2010, Set et al. [5] gave some new Hermite-Hadamard-type inequalities and other integral inequalities involving two functions. More details about results proved in [5] will be given in Section 2. In this paper, we present more general Hermite-Hadamard-type inequalities and some integral inequalities involving several functions.

Let and . The -norm of the function on is defined by

Below we recall few well-known inequalities that will be useful in the proofs of our results.

*Hermite-Hadamard’s Inequality (see [6–8]).* If is a convex function on , then
If is a concave function on , then

*Barnes-Gugunova-Levin Inequality (see [9–11]).* If and are nonnegative concave functions on , and if , then
where

*The Power-Mean Inequality (see [12]).* Let and let . Then
Notice that if then

*A Generalization of Hölder Integral Inequality.* For any , , if are nonnegative functions on and if are integrable functions on , then

*A Generalization of Minkowski Integral Inequality.* If and if are non-negative functions on such that for all , then

*A Generalization of Young Inequality.* If and , , then

To prove results, we refer to the following lemma.

Lemma 1 (see [13]). *If and if and are positive functions on such that for all , then
*

*Proof. *Let . Assume that and are positive functions on such that for all . Then
Then
Thus,

#### 2. Main Results

We start this section with the following.

Theorem 2. * Let be a positive even integer and and let be non-negative functions on such that are concave on . Then
**
where
**
for all . Moreover, if , then
*

*Proof. *Applying the inequality (3) with , for any , we get
and, consequently,

By the Barnes-Gudunova-Levin inequality (4), it follows that
where
for all .

By the power-mean inequality (7), we have
for all .

This implies the inequality (15).

Next, we assume that . By the inequality (19) and the generalized Hölder inequality, we obtain that

This proof is completed.

It is easy to notice that if we put in Theorem 2 then we get the following.

Corollary 3 (see [5]). * Let and let be non-negative functions on such that are concave on . Then
**
where
**
Moreover, if , then
*

Theorem 4. *Let and be a positive integer such that and let be positive functions on such that the functions are integrable functions on , for all , and
**
for all and for all . Then
**
where
**
for all .*

*Proof. * By Lemma 1, we have
for all .

Then
for all .

Let , , and for all .

It follows that
for all .

By multiplying the above inequalities and the generalized Minkowski inequality, we obtain that
Then
This implies the inequality (28).

Notice that from above theorem one can easily get the following.

Corollary 5 (see [5]). * Let and let be positive functions on such that , , and
**
for all . Then
**
where .*

*Proof. *By Theorem 4 where , we have
where

Let . Then
This implies the inequality (36).

Theorem 6. * Let be a positive integer such that and and let be non-negative functions on such that are concave on . Then
*

*Proof. *Using the inequality (3) with , for any , we obtain

Then

By the power-mean inequality (7), we have
so
for all .

Then
This implies the inequality (40).

One can easily check that if we put in Theorem 6 then we get the following.

Corollary 7 (see [5]). *Let and let be non-negative functions on such that are concave on . Then
*

Theorem 8. *Let be a positive integer such that and , , and let be positive functions on such that the function is integrable on , for all , and
**
for all and . Then
**
where
**
for all .*

*Proof. *By Lemma 1, we have
for all .

Using the elementary inequality where and , we get
for all .

Then
for all .

Let , , and
for all .

It follows that
for all .

By the generalized Young inequality, we obtain that
This proof is completed.

Applying Theorem 8 with and putting there , , and , we get the following.

Corollary 9 (see [5]). *Let , , , and let and be positive functions on such that , , and
**
for all . Then
**
where
*

#### Acknowledgment

The author would like to thank the referees for their useful comments and suggestions.

#### References

- B. G. Pachpatte, “On some inequalities for convex functions,”
*RGMIA Research Report Collection*, vol. 6, 9 pages, 2003. View at Google Scholar - B. G. Pachpatte, “A note on integral inequalities involving two log-convex functions,”
*Mathematical Inequalities & Applications*, vol. 7, no. 4, pp. 511–515, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - M. K. Bakula, M. E. Özdemir, and J. Pečarić, “Hadamard type inequalities for $m$-convex and $(\alpha ,m)$-convex functions,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 9, no. 4, article 96, 12 pages, 2008. View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. S. Mitrinović, J. E. Pečarić, and A. M. Fink,
*Classical and New Inequalities in Analysis*, vol. 61 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dodrecht, The Netherlands, 1993. View at MathSciNet - M. Alomari and M. Darus, “On the Hadamard's inequality for log-convex functions on the coordinates,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 283147, 13 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Dinu, “Hermite-Hadamard inequality on time scales,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 287947, 24 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Pečarić and T. Pejković, “On an integral inequality,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 5, no. 2, article 47, 6 pages, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. E. Pečarić, F. Proschan, and Y. L. Tong,
*Convex Functions, Partial Orderings, and Statistical Applications*, vol. 187 of*Mathematics in Science and Engineering*, 1992. View at MathSciNet - T. K. Pogány, “On an open problem of F. Qi,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 3, no. 4, article 54, 5 pages, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. S. Bullen, D. S. Mitrinović, and P. M. Vasić,
*Means and Their Inequalities*, vol. 31 of*Mathematics and Its Applications*, D. Reidel Publishing, Dordrecht, The Netherlands, 1988. View at MathSciNet - L. Bougoffa, “On Minkowski and Hardy integral inequalities,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 7, no. 2, article 60, 3 pages, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet