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

Emdat EGcan and Mehmet Kunt 1Department of Mathematics, Faculty of Sciences and Arts, Giresun University, 28200 Giresun, Turkey 2Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey Correspondence should be addressed to Mehmet Kunt; mkunt@ktu.edu.tr Received 16 July 2015; Accepted 24 January 2016 Academic Editor: S. T. Ali Copyright © 2016 İ. İşcan and M. Kunt.This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
Let  :  ⊆ R → R be a convex function defined on interval  of real numbers and ,  ∈  with  < .The inequality is 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 [10], Latif et al. established the following inequality which is the weighted generalization of Hermite-Hadamard inequality for GA-convex functions as follows.( 2

Journal of Mathematics
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  > 0 with  >  ≥ 0 are defined by respectively, where Γ() 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][13][14][15].
In [9], İs ¸can represented Hermite-Hadamard's inequalities for GA-convex functions in fractional integral forms as follows.
In [16], the authors represented Hermite-Hadamard-Fejér inequalities for GA-convex functions in fractional integral forms as follows.
and geometrically symmetric with respect to √ , then the following inequalities for fractional integrals hold: with  > 0.
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.

Main Results
where with  > 0.
Proof.From Lemma 8 we have Setting  =  1−   and  =  1−   ln(/) gives Since  : [, ] → R is geometrically symmetric with respect to √ , we write and then we have Since 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  = 1, one has the following Hermite-Hadamard-Fejér type inequality for quasi-geometrically convex function which is related to the right-hand side of ( 5 where with  > 0. Proof.Similar to the proof of Theorem 10, using Lemma 8, ( 14), (16) This completes the proof.

Corollary 13.
In Theorem 12, one has the following.
(1) If one takes  = 1, 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 () = 1, 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  = 1 and () = 1, one has the following Hermite-Hadamard type inequality for quasi-geometrically convex function: Here, we use for  ∈ [0, 1/2] and for  ∈ [1/2, 1], which follows from for any  ≥  ≥ 0 and  ≥ 1.Hence, inequality (28) is proved.

Corollary 15.
In Theorem 14, one can see the following.