Hermite-Hadamard Type Inequalities for (m, h1, h2)-Convex Functions Via Riemann-Liouville Fractional Integrals
1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, China
2Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, China
3Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China
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
Turkish Journal of Analysis and Number Theory, 2014 2 (1),
Received March 4, 2014; revised March 14, 2014; accepted March 16, 2014Copyright © 2013 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Shi, De-Ping, Bo-Yan Xi, and 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 (2014): 23-28.
- Shi, D. , Xi, B. , & Qi, F. (2014). 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.
- Shi, De-Ping, Bo-Yan Xi, and 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, no. 1 (2014): 23-28.
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The following definitions are well known in the literature.
Definition 1.1. A function is said to be convex if
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. () For , a function is said to be -convex if
holds for all and . If the above inequality (2) reverses, then is said to be -concave on .
Definition 1.3. () Let , be an interval, and . A function is said to be -convex if the inequality
If the above inequality (3) reverses, then is said to be -concave on .
Definition 1.4. () 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. () Let be a differenti-able function on , where and . If is a convex function on , then
Theorem 1.2. () Letis a differentiable function on ,is convex function on , where ，then
Theorem 1.3. () Letis a differentiable function on , where ，if is convex function on , then
Theorem 1.4. () Letis-convex function, where , if ，then
Definition 1.6. () Let ，The Riemann-Liouville integrals and of order with are defined by
Respectively, where and is the classical Euler gamma function be defined by
Theorem 1.5. () Letbe a positive function with , If is a convex function on then
Theorem 1.6. () 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 , 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
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
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
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.
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