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Hermite-Hadamard Type Inequalities for (m, h₁, h₂)-Convex Functions Via Riemann-Liouville Fractional Integrals

De-Ping Shi, Bo-Yan Xi*, Feng Qi

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

Abstract: In the paper, via Riemann-Liouville fractional integration, the authors present some new inequalities of Hermite-Hadamard type for functions whose derivatives in absolute value are (m, h₁, h₂)-convex.

Keywords: Riemann-Liouville fractional integral, (m, h1, h2)-convex function, integral inequality of Hermite-Hadamard type

Received March 4, 2014; revised March 14, 2014; accepted March 16, 2014

Citation: De-Ping Shi, Bo-Yan Xi, Feng Qi, Hermite-Hadamard Type Inequalities for (m, h₁, h₂)-Convex Functions Via Riemann-Liouville Fractional Integrals. Turkish Journal of Analysis and Number Theory, 2 (1). 23-28. January 2014. doi:10.12691/tjant-2-1-6

1. Introduction

The following definitions are well known in the literature.

Definition 1.1. A function is said to be convex if

[picture] | (1)

holds for all and . If the inequality (1) reverses, then is said to be concave on .

The well-known Hermite-Hadamard inequality reads that for every convex function ,we have

, where with . If is concave, the above inequalities reverse.

Definition 1.2. ([2]) For , a function is said to be -convex if

[picture] | (2)

holds for all and . If the above inequality (2) reverses, then is said to be -concave on .

Definition 1.3. ([6]) Let , be an interval, and . A function is said to be -convex if the inequality

[picture] | (3)

If the above inequality (3) reverses, then is said to be -concave on .

Definition 1.4. ([10]) For, if

is valid for all and ,then we say that is a -convex function on .

There have been many inequalities of Hermite-Hadamard type for the above convex functions. Some of them may be recited as follows.

Theorem 1.1. ([2]) Let be a differenti-able function on , where and . If is a convex function on , then

Theorem 1.2. ([3]) Letis a differentiable function on ,is convex function on , where ，then

Theorem 1.3. ([4]) Letis a differentiable function on , where ，if is convex function on , then

Theorem 1.4. ([5]) Letis-convex function, where , if ，then

Definition 1.6. ([1]) Let ，The Riemann-Liouville integrals and of order with are defined by

and

Respectively, where and is the classical Euler gamma function be defined by

Theorem 1.5. ([12]) Letbe a positive function with , If is a convex function on then

with .

Theorem 1.6. ([12]) Let be a differentiable mapping on with , If is convex on , then

In this paper, motivated by the above results, we will establish a Riemann-Liouville fractional integral identity involving a differentiable mapping and present some new inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals for (m, h₁, h₂)-convex functions.

2. A Definition and A Lemma

In the most recent paper [11], Maksa and Palés introduced even more general notion of convexity. More precisely, -convex functions are defined as solutions f of the functional inequality

where and are given functions.

We first introduce a definition of -convex functions.

Definition 2.1. Assume, and .Then is said to be (m, h₁, h₂)-convex if the inequality

holds for all and . If the above inequality reverses, then is said to be (m, h₁, h₂)-concave on .

Let be a differentiable function on and . Denote by

Specially, when , we have

Lemma 2.1. Let be a differentiable function on such that . Then

Proof. Letting. By integration by parts, we have

Similarly, we obtain

and

The proof of Lemma 2.1 is complete.

3. New Inequalities for (m, h₁, h₂)-Convex Functions

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

where for and is the classical Beta function which may be defined by

Proof. From Lemma 2.1, Hölder inequality, and the (m, h₁, h₂)-convexity of , we obtain

where

and

The proof of Theorem 3.1 is complete.

Corollary 3.1.1. Under the conditions of Theorem 3.1, if then

Furthermore, if then

Corollary 3.1.2. Under the conditions of Corollary 3.1.1, if then

Specially, if , then

Corollary 3.1.3. Under the conditions of Theorem 3.1,if and , then

Specially, if then

Theorem 3.2. Let be a differentiable function such that and . If is -convex on for and , then

where where is as given in Theorem 3.1.

Proof. From Lemma 2.1, Hölder inequality, and the -convexity of , we obtain

The proof of Theorem 3.2 is complete.

Corollary 3.2.1. Under the conditions of Theorem 3.2, (1) if and , then

(2) if and , then

(3) if and , then

(4) if and , then

Acknowledgement

This work was partially supported by the NNSF under Grant No. 11361038 of China and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159, China.

References

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