#### 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 [1], 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 [2]).

In recent years, many authors have studied errors estimations for Hermite-Hadamard, Ostrowski, and Simpson inequalities; for refinements, counterparts, and generalization see [3–10].

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

*Definition 7 (see [13]). *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 [13–19].

Lemma 8 (see [20]). *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 [21], 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 [21], 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.