/ / Article

Research Article | Open Access

Volume 2016 |Article ID 4361806 | 8 pages | https://doi.org/10.1155/2016/4361806

# Some New Generalized Integral Inequalities for GA--Convex Functions via Hadamard Fractional Integrals

Revised14 Jul 2016
Accepted01 Aug 2016
Published01 Sep 2016

#### Abstract

We prove new generalization of Hadamard, Ostrowski, and Simpson inequalities in the framework of GA--convex functions and Hadamard fractional integral.

#### 1. Introduction

Let a real function be defined on a nonempty interval of real line . The function is said to be convex on if inequalityholds for all and

In , Breckner introduced -convex functions as a generalization of convex functions as follows.

Definition 1. Let be a fixed real number. A function is said to be -convex (in the second sense), or that belongs to the class , if for all and .

Of course, -convexity means just convexity when .

The following inequalities are well known in the literature as Hermite-Hadamard inequality, Ostrowski inequality, and Simpson inequality, respectively.

Theorem 2. Let be a convex function defined on the interval of real numbers and with . The following double inequality holds:

Theorem 3. Let be a mapping differentiable in , the interior of I, and let with If , , then the following inequality holds: for all

Theorem 4. Let be a four times’ continuously differentiable mapping on and Then the following inequality holds:

We will give definitions of the right and left hand side Hadamard fractional integrals which are used throughout this paper.

Definition 5. Let . The right-sided and left-sided Hadamard fractional integrals and of order with are defined byrespectively, where is the Gamma function defined by (see ).

In recent years, many authors have studied errors estimations for Hermite-Hadamard, Ostrowski, and Simpson inequalities; for refinements, counterparts, and generalization see .

Definition 6 (see [11, 12]). A function is said to be GA-convex (geometric-arithmetically convex) if for all and .

Definition 7 (see ). For , a function is said to be GA--convex (geometric-arithmetically -convex) if for all and .

It can be easily seen that if , GA--convexity reduces to GA-convexity.

For recent results and generalizations concerning GA-convex and GA--convex functions see .

Lemma 8 (see ). For and , one has where

Let be a differentiable function on , the interior of ; in sequel of this paper we will take where with , , , , and is Euler Gamma function.

In , can gave Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms as follows.

Theorem 9. Let be a function such that , where with . If is a GA-convex function on , then the following inequalities for fractional integrals hold: with .

In , can obtained some new inequalities for quasi-geometrically convex functions via fractional integrals by using the following lemma.

Lemma 10. Let be a differentiable function on such that , where with . Then for all , , and one has

In this paper, we will use Lemma 10 to obtain some new inequalities on generalization of Hadamard, Ostrowski, and Simpson type inequalities for GA--convex functions via Hadamard fractional integral.

#### 2. Generalized Integral Inequalities for Some GA--Convex Functions via Fractional Integrals

Theorem 11. Let be a differentiable function on such that , where with . If is GA--convex on in the second sense for some fixed , , , and then the following inequality for fractional integrals holds:where

Proof. Using Lemma 10, property of the modulus, and the power-mean inequality, we have Since is GA--convex on , we getand by a simple computation, we haveHence, If we use (18), (19), and (20) in (17), we obtain the desired result. This completes the proof.

Corollary 12. Under the assumptions of Theorem 11 with , inequality (15) reduces to the following inequality:

Corollary 13. Under the assumptions of Theorem 11 with and , inequality (15) reduces to the following inequality: where

Corollary 14. Under the assumptions of Theorem 11 with , inequality (15) reduces to the following inequality:

Corollary 15. Under the assumptions of Theorem 11 with , from inequality (15), one gets the following Simpson type inequality for fractional integrals:

Corollary 16. Under the assumptions of Theorem 11 with , from inequality (15), one gets

Corollary 17. Under the assumptions of Theorem 11 with and , from inequality (15) one gets

Corollary 18. Let the assumptions of Theorem 11 hold. If for all and , then from inequality (15), one gets the following Ostrowski type inequality for fractional integrals: for all

Theorem 19. Let be a differentiable function on such that , where with . If is GA--convex on for some fixed , , , and then the following inequality for fractional integrals holds: where and

Proof. Using Lemma 10, property of the modulus, the Hölder inequality, and GA--convexity of , we have where , andUsing Lemma 8, we have Hence, if we use (32)-(33) in (31) and replacing , we obtain the desired result. This completes the proof.

Corollary 20. Under the assumptions of Theorem 19 with , inequality (29) reduces to the following inequality:

Corollary 21. Under the assumptions of Theorem 19 with and , inequality (29) reduces to the following inequality:

Corollary 22. Under the assumptions of Theorem 19 with , from inequality (29), one gets the following Simpson type inequality for fractional integrals:

Corollary 23. Under the assumptions of Theorem 19 with , from inequality (29), one gets

Corollary 24. Under the assumptions of Theorem 19 with and , from inequality (29) one gets

Corollary 25. Let the assumptions of Theorem 19 hold. If for all and , then from inequality (29), one gets the following Ostrowski type inequality for fractional integrals: for each

#### Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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