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

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 [1–3] 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 [9–23].

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

The weighted geometric mean of data with weight is defined by

The generalized logarithmic mean of data is defined by

The identric mean of data is defined by

Applying Corollary 7 with on , we get the following:

Applying Corollary 9 with on , we get the following:

Applying Corollary 7 with on , we get the following:

Applying Corollary 9 with on , we get the following:

Applying Corollary 14 with on , we get the following:

Applying Corollary 16 with on , we get the following:

Applying Corollary 14 with on , we get the following:

Applying Corollary 16 with on , we get the following:

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

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