`The Scientific World JournalVolume 2014 (2014), Article ID 717164, 13 pageshttp://dx.doi.org/10.1155/2014/717164`
Research Article

Generalizations on Some Hermite-Hadamard Type Inequalities for Differentiable Convex Functions with Applications to Weighted Means

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

Received 5 August 2013; Accepted 10 October 2013; Published 16 January 2014

Academic Editors: J.-S. Chen and T. Li

Copyright © 2014 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

Some new Hermite-Hadamard type inequalities for differentiable convex functions were presented by Xi and Qi. In this paper, we present new generalizations on the Xi-Qi inequalities.

1. Introduction

The Hermite-Hadamard inequality [13] states that if is a convex function on , then

Let be differentiable on and with . Below we recall some Hermite-Hadamard type inequalities.

In 1998, Dragomir and Agarwal [4] showed that (i) if is convex on , then

and (ii) if is convex on with , then

In 2000, Pearce and Pečarić [5] showed that if is convex on with , then

In 2004, Kirmaci [6] showed that if is convex on with , then

In 2010, Sarikaya et al. [7] showed that if and is convex on with , then

In 2012, Xi and Qi [8] showed that if , and if and is convex on with , then

Moreover, for other results involving the Hermite-Hadamard type inequalities, we also refer to [923].

In this paper, we generalize the Xi-Qi inequalities.

2. Preliminaries

Lemma 1. Let ,   and let be differentiable on and with . Assume that and . Then

Proof. Integrating by part and changing variable, we have
Thus,

Lemma 2 (see [8]). Let and . Then

3. Main Results

Theorem 3. Let ,   and let be differentiable on and with . Assume that and . If is convex on with , then

Proof. Suppose that is convex on with . By Lemma 1, we have
Case  . By the convexity of and Lemma 2, we have Thus,
Case  . By Hölder's inequality, we have
By the convexity of and Lemma 2, we have Thus, This proof is completed.

It is easy to notice that if we put in Theorem 3 then we get the following.

Corollary 4 (see [8]). Let ,   and let be differentiable on and with . Assume that . If is convex on with , then

One can easily check that if we put in Theorem 3, then we get the following.

Corollary 5. Let ,   and let be differentiable on and with . Assume that . If is convex on with , then

One can easily check that if we put in Theorem 3 then we get the following.

Corollary 6. Let ,   and let be differentiable on and with . Assume that . If is convex on with , then

It is easy to notice that if we put in Theorem 3 then we get the following.

Corollary 7. Let be differentiable on and with . Assume that and . If is convex on with , then

It is easy to notice that if we put in Theorem 3 then we get the following.

Corollary 8. Let be differentiable on and with . Assume that    and . If is convex on with , then

It is easy to notice that if we put in Theorem 3 then we get the following.

Corollary 9. Let be differentiable on and with . Assume that and . If is convex on with , then

Theorem 10. Let ,   and let be differentiable on and with . Assume that and . If is convex on with , then

Proof. Suppose that is convex on with . If , then, by Theorem 3, we have
Next, we suppose that . By Lemma 1 and Hölder’s inequality, we have
By the convexity of and Lemma 2, we have Thus,
This proof is completed.

It is easy to notice that if we put in Theorem 10 then we get the following.

Corollary 11 (see [8]). Let ,   and let be differentiable on and with . Assume that . If is convex on with , then

One can easily check that if we put in Theorem 10 then we get the following.

Corollary 12. Let ,   and let be differentiable on and with . Assume that . If is convex on with , then

One can easily check that if we put in Theorem 10 then we get the following.

Corollary 13. Let ,   and let be differentiable on and with . Assume that   . If is convex on with , then

It is easy to notice that if we put in Theorem 10 then we get the following.

Corollary 14. Let be differentiable on and with . Assume that and . If is convex on with , then

It is easy to notice that if we put in Theorem 10 then we get the following.

Corollary 15. Let be differentiable on and with . Assume that and . If is convex on with , then

It is easy to notice that if we put in Theorem 10 then we get the following.

Corollary 16. Let be differentiable on and with . Assume that and . If is convex on with , then

4. Applications

In this section, we suppose that and with . Let .

The weighted arithmetic mean of data with weight is defined by