Journal of Function Spaces and Applications

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

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

Research Article

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

Banyat Sroysang

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.

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