Research Article | Open Access

# Applying GG-Convex Function to Hermite-Hadamard Inequalities Involving Hadamard Fractional Integrals

**Academic Editor:**Hari M. Srivastava

#### Abstract

By virtue of fractional integral identities, incomplete beta function, useful series, and inequalities, we apply the concept of GG-convex function to derive new type Hermite-Hadamard inequalities involving Hadamard fractional integrals. Finally, some applications to special means of real numbers are demonstrated.

#### 1. Introduction

Fractional calculus played an important role in various fields such as electricity, biology, economics, and signal and image processing [1â€“8]. The fractional Hermite-Hadamard inequality gives a lower and an upper estimation for both right-hand and left-hand integrals average of any convex function defined on a compact interval, involving the midpoint and the endpoints of the domain.

As we know, Set [9] firstly studied fractional Ostrowski inequalities involving Riemann-Liouville fractional integrals. Then, Sarikaya et al. [10] studied Hermite-Hadamard type inequalities involving Riemann-Liouville fractional integrals. Further, our group go on studying fractional version Hermite-Hadamard inequality involving Riemann-Liouville and Hadamard fractional integrals for all kinds of functions [11â€“19].

Recently, Wang et al. [16, 17] established the following two powerful fractional integral identities involving Hadamard fractional integrals.

Lemma 1 (see [12, Lemma 3.1]). *Let be a differentiable mapping on . If , then the following equality for fractional integrals holds:**
where the symbols and are defined by
**
where is the Gamma function.*

Lemma 2 (see [11, Lemma 2.1]). *Let be a differentiable mapping on with . If , then the following equality for fractional integrals holds:**
where
*

*Remark 3. *It is remarkable that Professor Srivastava et al. [20] give some further refinements and extensions of the Hermite-Hadamard inequalities in variables. In the forthcoming works, we will try to extend to study fractional version Hermite-Hadamard inequalities in variables based on such fundamental results.

Next, we recall the following basic concepts and results in our previous papers.

*Definition 4 (see [21, 22]). *Let . A function is said to be GG-convex on if, for every and , one has

*Remark 5. *By the arithmetic-geometric mean inequality, we have
Linking (5) and (6), we obtain
which appears in the standard definition of GA-convex function [21]. So GG-convex function is GA-convex function.

Lemma 6 (see [19, Lemma 2.5]). *For , one has
*

Lemma 7 (see [13, Lemma 2.1]). *For and , one has
**
where .*

Lemma 8 (see [13, Lemma 2.2]). *For and , one has
*

Lemma 9 (see [13, Lemma 2.3]). *For and , one has
*

In the present paper, we will use the above concepts and lemmas to derive some new fractional Hermite-Hadamard inequalities involving Hadamard fractional integrals.

#### 2. Main Results Based on Lemma 1

Now we are ready to state the following main results in this section.

Theorem 10. *Let be a differentiable mapping. If is measurable and is GG-convex on for some fixed and , , then the following integrals hold:
*

*Proof. *Noting Definition 4 and Lemmas 1 and 6, we have
The proof is done.

Theorem 11. *Let be a differentiable mapping. If is measurable and , is GG-convex on for some fired and , , then the following integrals hold:**
where .*

*Proof. *By using Definition 4 and Lemmas 1 and 6, we have
The proof is done.

Theorem 12. *Let be a differentiable mapping. If is measurable and is GG-convex on for some fired , and , , then the following integrals hold:
*

*Proof. *By using Definition 4 and Lemmas 1, 7, 8, and 9, we have
The proof is done.

Theorem 13. *Let be a differentiable mapping. If is measurable and , is GG-convex on , for some fired , , and , , then the following integrals hold:
**
where .*

*Proof. *By using Definition 4 and Lemmas 1, 7, 8, and 9, we have
The proof is done.

#### 3. Main Results Based on Lemma 2

Theorem 14. *Let be a differentiable mapping. If is measurable and is GG-convex on , for some fired and , , then the following integrals hold:
*