Conformable Fractional Integrals Versions of Hermite-Hadamard Inequalities and Their Generalizations

Copyright © 2018 Muhammad Adil Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, s-convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized HermiteHadamard type inequalities.


Introduction
The class of convex functions is well known in the literature and is usually defined in the following way: let I be an interval in R; then a function  : I → R is said to be convex on I if the inequality holds for all ,  ∈ I and  ∈ [0, 1].Also, we say that  is concave, if the inequality in (1) holds in the reverse direction.
There are several generalizations of the convex function.
Here we mention basic definition of -convex function and coordinate convex function.In the paper [1], Hudzik and Maligranda considered a generalization of convex function, which is known as -convex function in the second sense.This class of function is defined in the following way: a function  : [0, +∞) → R is said to be -convex in the second sense if holds for all ,  ≥ 0 and  ∈ [0, 1] and for some fixed  ∈ (0, 1].The class of -convex functions in the second sense is usually denoted by  2  .
In [2], the concept of convex functions defined on the coordinates of the bidimensional interval of the plane of two variables was introduced.Remark 2. Note that every convex function  : [, ] × [, ] → R is convex on the coordinates, but the converse is not generally true [2].
Many important inequalities have been obtained for this class of functions but here we will present only one of them.
If  : I → R is a convex function on the interval I, then, for any ,  ∈ I with  ̸ = , we have the following double inequality: Both inequalities hold in reverse direction if the function  is concave on the interval I.This remarkable result was given in ( [3], 1893) and is well known in the literature as Hermite-Hadamard inequality.Since its discovery, this inequality has become the center of interest for many prolific researchers and received a considerable attention.Also, a number of extensions, generalizations, and variants of (3) have been provided in the theory of mathematical inequalities.For example, see [4][5][6][7][8][9][10][11][12] and the references cited therein.Now we recall some definitions and important results in the theory of conformable fractional calculus.For detailed treatment of the results, we refer the interested readers to [13][14][15][16][17][18][19][20].
Several important variants of Hermite-Hadamard inequality have been provided in the literature, such as the versions established by Anderson [21] and Sarikaya et al. [26] and so forth.
In this paper, we prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, -convex, and coordinate convex functions.We prove new Montgomery identity for conformable fractional integral.By using this identity, we obtain Hermite-Hadamard type inequalities.These results give us the generalizations of the earlier results.

Hermite-Hadamard Inequalities
Theorem 12. Let  ∈ (0, 1],  : [, ] → [0, +∞) be a convex function defined on [, ], where 0 <  < ; then the following double inequality holds: Proof.Let us define a function  on [, ] by Obviously the function  is increasing and continuous function on [, ].Therefore, and hence Now (by using (17)) . ( By changing of variable and convexity of , we get Hence, Now let us define a function ℎ on [, ] by Clearly the function ℎ is decreasing and continuous on [, ].Therefore, and hence Now Hence, From ( 20) and ( 25), we deduce the right-hand side of ( 14).Now we prove left inequality in (14).
It is well known that Also from the functions  and ℎ as defined in ( 15) and ( 21), respectively, we have Therefore, By using (28) in (26), we obtain Now, by changing of variable and using the fact that −1 Similarly, Now Combining ( 29), (30), (31), and (32), we get which is equivalent to the left inequality in (14).
Corollary 13.Under the assumptions of Theorem 12, if we put  = 1, we get the following well-known Hermite-Hadamard inequality for convex function: Now we prove Hermite-Hadamard inequality for conformable fractional integral by using -convex function.
Corollary 15.Under the assumptions of Theorem 14, if we put  = 1, we get the following well-known Hermite-Hadamard inequality for -convex function [27]: In the following theorem, we prove Hermite-Hadamard inequality for conformable fractional integral by using coordinate convex function.

Generalization of Hermite-Hadamard Type Inequalities
Now, we are in position to find some new estimations for the left-hand side of Hermite-