Abstract

In this paper, some new inequalities of the trapezoid type for h-convex functions via generalized fractional integral are given. The results also provide new estimates on these types of trapezoid inequalities for Riemann-Liouville type fractional operators.

1. Introduction

First, we recall some necessary definitions and mathematical preliminaries of the generalized fractional integrals which are defined by Sarikaya and Ertugral ¹.

Let which satisfies the following condition:

We define the following left-sided and right-sided generalized fractional integral operators, respectively, as follows:

(1.1)

(1.2)

The most important feature of generalized fractional integrals is that they generalize some types of fractional integrals such as Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral, Katugampola fractional integrals, conformable fractional integral, Hadamard fractional integrals, etc. These important special cases of the integral operators (1.1) and (1.2) are mentioned below.

a) If we take the operator (1.1) and (1.2) reduce to the Riemann integral as follows:

b) If we take the operator (1.1) and (1.2) reduce to the Riemann-Liouville fractional integral as follows:

c) If we take the operator (1.1) and (1.2) reduce to the k-Riemann-Liouville fractional integral as follows:

where

and

are given by Mubeen and Habibullah in ².

Recently, in ¹, Sarikaya and Ertugral established the following Trapezoid inequalities for generalized fractional integrals:

Theorem 1.1. Let be a differentiable mapping on with . If is convex on then the following inequality for generalized fractional integrals holds:

where

Theorem 1.2. Let be a differentiable mapping on with If is convex on then the following inequality for generalized fractional integrals holds:

Recently, in ³, Ertugral and Sarikaya established the following Trapezoid inequalities for generalized fractional integrals:

Theorem 1.3. Let be an absolutely continuous mapping on such that where with . If the mapping is

Theorem 1.4. Let be a differentiable mapping on with If the mapping

where .

In ⁴, Varošanec introduced the following class of functions.

Definition 1.1. Let be a positive function. We say that is -convex, or that belongs to the class , if is nonnegative and for all and we have

(1.3)

If inequality (1.3) is reversed, then is said to be -concave, i.e. .

The systematic study of h-convex functions with their various applications has been given by many authors, see ^{6, 7, 8, 9, 10}.

In this paper, we establish some trapezoid type inequalities via generalized fractional integrals for h-convex functions.

2. New Integral Inequalities via P-preinvexity

For our results, we need the following important fractional integrtal identity ³:

Lemma 2.1. Let be an absolutely continuous mapping on such that , where with . Then the following equality holds:

(2.1)

where

and

Theorem 2.1. Let be an absolutely continuous mapping on such that where with If the mapping is h-convex on then we have the following inequality

Proof. By Lemma 2.1 and h-convexity of on we have

this is the required result.

Remark 2.1. Under assumptions of Theorem 2.1, if then Theorem 2.1 reduces to Theorem 1 in ³.

Remark 2.2. Under assumptions of Theorem 2.1, if then Theorem 2.1 reduces to Theorem 4 in ⁵.

Corollary 2.1. Under assumptions of Theorem 2.1,

(1) if and , then

(2) if and then

Remark 2.3. Under assumptions of Theorem 2.1,

(1) if and then Theorem 2.1 reduces to Corollary 2 in ³.

(2) if and then Theorem 2.1 reduces to Corollary 3 in ³.

Theorem 2.2. Let be an absolutely continuous mapping on such that where with If the mapping is h-convex on then we have the following inequality

Proof. Using the h-convexity of on Lemma 2.1, and Hölder's inequality, we have

where we have used the fact that

Therefore

Remark 2.4. Under assumptions of Theorem 2.2, if then Theorem 2.2 reduces to Theorem 2 in ³.

Remark 2.5. Under assumptions of Theorem 2.2, if then Theorem 2.2 reduces to Theorem 5 in ⁵.

Corollary 2.2. Under assumptions of Theorem 2.2, if and then

Remark 2.6. Under assumptions of Theorem 2.2, and then Theorem 2.2 reduce to Corollary 4 in ³.

References