Hermite-Hadamard-Fejér Type Inequalities for Quasi-Geometrically Convex Functions via Fractional Integrals

Abstract

Some Hermite-Hadamard-Fejér type integral inequalities for quasi-geometrically convex functions in fractional integral forms have been obtained.

1. Introduction

Let be a convex function defined on interval of real numbers and with . The inequalityis well known in the literature as Hermite-Hadamard’s inequality [1].

The most well-known inequalities related to the integral mean of a convex function are the Hermite-Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fejér inequalities.

In [2], Fejér established the following Fejér inequality which is the weighted generalization of Hermite-Hadamard inequality (1).

Theorem 1. Let be convex function. Then, the inequalityholds, where is nonnegative, integrable, and symmetric to .

For some results which generalize, improve, and extend inequalities (1) and (2), see [3–6].

Definition 2 (see [7, 8]). A function is said to be GA-convex (geometric-arithmetically convex) if for all and .

Definition 3 (see [9]). A function is said to be quasi-geometrically convex on if for all and .

In [10], Latif et al. established the following inequality which is the weighted generalization of Hermite-Hadamard inequality for GA-convex functions as follows.

Theorem 4. Let be a GA-convex function and with . Let be continuous positive mapping and geometrically symmetric to . Then,

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

Definition 5 (see [11]). Let . The right-hand side and left-hand side Hadamard fractional integrals and of order with are defined by respectively, where is the Gamma function defined by .

Because of the wide application of Hermite-Hadamard type inequalities and fractional integrals, many researchers extend their studies to Hermite-Hadamard type inequalities involving fractional integrals not limited to integer integrals. Recently, more and more Hermite-Hadamard inequalities involving fractional integrals have been obtained for different classes of functions; see [9, 12–15].

In [9], İşcan represented Hermite-Hadamard’s inequalities for GA-convex functions in fractional integral forms as follows.

Theorem 6. 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 [16], the authors represented Hermite-Hadamard-Fejér inequalities for GA-convex functions in fractional integral forms as follows.

Theorem 7. Let be a GA-convex function with and . If is nonnegative, integrable, and geometrically symmetric with respect to , then the following inequalities for fractional integrals hold:with .

Lemma 8 (see [16]). Let be a differentiable mapping on with and . If is integrable and geometrically symmetric with respect to , then the following equality for fractional integrals holds:with .

Lemma 9 (see [15, 17]). For and , one has

In [9], İşcan introduced the quasi-geometrically convex functions and presented some Hermite-Hadamard type integral inequalities for quasi-geometrically convex functions via fractional integrals. In [18], İşcan obtained generalized Hermite-Hadamard type integral inequalities for quasi-geometrically convex functions. In [19], İşcan et al. showed some Hermite-Hadamard and Simpson type inequalities for differentiable quasi-geometrically convex functions.

In this paper, it is the first time Hermite-Hadamard-Fejér type integral inequality for quasi-geometrically convex function has been studied. Obtained results are new and generalize the Hermite-Hadamard type integral inequality for quasi-geometrically convex functions.

2. Main Results

Throughout this section, let , for the continuous function .

Theorem 10. Let be a differentiable mapping on and with . If is quasi-geometrically convex on and is continuous and geometrically symmetric with respect to , then the following inequality for fractional integrals holds:wherewith .

Proof. From Lemma 8 we haveSetting and givesSince is geometrically symmetric with respect to , we writeand then we haveSince is quasi-geometrically convex on , we know that, for ,and a combination of (14), (16), and (17) gives This completes the proof.

Corollary 11. In Theorem 10, one can see the following.
(1) If one takes , one has the following Hermite-Hadamard-Fejér type inequality for quasi-geometrically convex function which is related to the right-hand side of (5): (2) If one takes , one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function in fractional integral forms which is related to the right-hand side of (7):(3) If one takes and , one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function:

Theorem 12. Let be a differentiable mapping on and with . If , , is quasi-geometrically convex on and is continuous and geometrically symmetric with respect to , then the following inequalities for fractional integrals hold: wherewith .

Proof. Similar to the proof of Theorem 10, using Lemma 8, (14), (16), power mean inequality, and quasi-geometrically convexity of , we have This completes the proof.

Corollary 13. In Theorem 12, one has the following.
(1) If one takes , one has the following Hermite-Hadamard-Fejér type inequality for quasi-geometrically convex function which is related to the right-hand side of (5): (2) If one takes , one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function in fractional integral forms which is related to the right-hand side of (7): (3) If one takes and , one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function:

Theorem 14. Let be a differentiable mapping on and with . If , , is quasi-geometrically convex on and is continuous and geometrically symmetric with respect to , then the following inequalities for fractional integrals hold:
(i)with ;
(ii)for , where .

Proof. (i) Using Lemma 8, (14), (16), Hölder’s inequality, and quasi-geometrically convexity of , we haveHere, we use for and for , which follows from for any and . Hence, inequality (28) is proved.
(ii) Inequality (29) is easily proved using (30) and Lemma 9.

Corollary 15. In Theorem 14, one can see the following.
(i) In (28), consider the following:(1)If one takes , one has the following Hermite-Hadamard-Fejér type inequality for quasi-geometrically convex function which is related to the right-hand side of (5): (2) If one takes , one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function in fractional integral forms which is related to the right-hand side of (7): (3) If one takes and , one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function: (ii) In (29), consider the following:(1) If one takes, one has the following Hermite-Hadamard-Fejér type inequality for quasi-geometrically convex function which is related to the right-hand side of (5): (2) If one takes , one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function in fractional integral forms which is related to the right-hand side of (7): (3) If one takes and , one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function:

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.