Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 980438, 14 pages

doi:10.1155/2012/980438

Research Article

Some Integral Inequalities of Hermite-Hadamard Type for Convex Functions with Applications to Means

Bo-Yan Xi¹ and Feng Qi2,3

¹College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City 028043, China

²School of Mathematics and Informatics, Henan Polytechnic University, Henan Province, Jiaozuo City 454010, China

³Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City 300387, China

Received 21 February 2012; Accepted 21 May 2012

Academic Editor: Lars Diening

Copyright © 2012 Bo-Yan Xi and Feng Qi. 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.

The authors establish some new inequalities for differentiable convex functions, which are similar to the celebrated Hermite-Hadamard's integral inequality for convex functions, and apply these inequalities to construct inequalities for special means of two positive numbers.

1. Introduction

In [1], the following Hermite-Hadamard type inequalities for differentiable convex functions were proved.

Theorem 1.1 (see [1, Theorem 2.2]). Let be a differentiable mapping on ,   with . If is convex on , then

Theorem 1.2 (see [1, Theorem 2.3]). Let be a differentiable mapping on , with , and let . If the new mapping is convex on , then

In [2], the above inequalities were generalized as follows.

Theorem 1.3 (see [2, Theorems 1 and 2]). Let be differentiable on , with , and let . If is convex on , then

In [3], the above inequalities were further generalized as follows.

Theorem 1.4 (see [3, Theorems 2.3 and 2.4]). Let be differentiable on , with , and let . If is convex on , then

In [4], an inequality similar to the above ones was given as follows.

Theorem 1.5 (see [4, Theorem 3]). Let be an absolutely continuous mapping on whose derivative belongs to . Then

where and .

Recently, the following inequalities were obtained in [5].

Theorem 1.6. Let be differentiable on , with , and . If for is convex on , then

In this paper, we will establish some new Hermite-Hadamard type integral inequalities for differentiable functions and apply them to derive some inequalities of special means.

2. Lemmas

For establishing new integral inequalities of Hermite-Hadamard type, we need the lemmas below.

Lemma 2.1. Let be differentiable on , with . If and , then

Proof. Integrating by part and changing variable of definite integral yield

Similarly, we have

Adding these two equations leads to Lemma 2.1.

Lemma 2.2. For and , one has

Proof. This follows from a straightforward computation of definite integrals.

3. Some Integral Inequalities of Hermite-Hadamard Type

Now we are in a position to establish some new integral inequalities of Hermite-Hadamard type for differentiable convex functions.

The first main result is Theorem 3.1.

Theorem 3.1. Let be a differentiable function on , with , , and . If for is convex on , then

Proof. For , by Lemma 2.1, the convexity of on , and the noted Hölder's integral inequality, we have

In virtue of Lemma 2.2, a direct calculation yields

Substituting the above two equalities into the inequality (3.2) and utilizing Lemma 2.2 result in the inequality (3.1) for .

For , from Lemmas 2.1 and 2.2 it follows that

which is just equivalent to (3.1) for . Theorem 3.1 is proved.

If taking in Theorem 3.1, we derive the following corollary.

Corollary 3.2. Let be differentiable on , with , , and . If is convex on for , then

If letting , respectively, in Theorem 3.1, we can deduce the inequalities below.

Corollary 3.3. Let be differentiable on , with , and . If is convex on for , then

If setting in Corollary 3.3, then one has the following.

Corollary 3.4. Let be differentiable on , with , and . If is convex on , then

The second main result is Theorem 3.5.

Theorem 3.5. Let be differentiable on , with , 0 ≤ λ, μ  ≤ 1, and . If is convex on for , then

Proof. For , by the convexity of on , Lemma 2.1, and Hölder's integral inequality, it follows that

By Lemma 2.2, we have

Substituting these two equalities into the inequality (3.9) yields (3.8) for .

For , the proof is the same as the deduction of (3.4). Thus, Theorem 3.5 is proved.

As the derivation of corollaries of Theorem 3.1, we can obtain the following corollaries of Theorem 3.5.

Corollary 3.6. Let be differentiable on , with , , and . If is convex for on , then

Corollary 3.7. Let be differentiable on , with , and . If is convex for on , then

Corollary 3.8. Let be differentiable on , with , and . If is convex on , then

Corollary 3.9. Let be differentiable on , with , and . If is convex for on and

then

In particular, when , one has

4. Applications to Means

For two positive numbers and , define

for . These means are, respectively, called the arithmetic, geometric, harmonic, generalized logarithmic, identric, and Heronian means of two positive number and .

Applying Theorem 3.1 to for and leads to the following inequalities for means.

Theorem 4.1. Let , , either and or , then

In particular, if or , then

Taking for in Theorem 3.1 results in the following inequalities for means.

Theorem 4.2. For and , one has

In particular,

Finally, we can establish an inequality for Heronian mean as follows.

Theorem 4.3. For , , and or , one has

Proof. Let for and . Then

In virtue of Corollary 3.3, it follows that

On the other hand, we have

The proof is complete.

Remark 4.4. Some inequalities of Hermite-Hadamard type were also obtained in [6–9] by the authors.

Acknowledgment

The first author was supported in part by the National Natural Science Foundation of China under Grant no. 10962004.

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