Estimates of Upper Bound for a Function Associated with Riemann-Liouville Fractional Integral via h-Convex Functions

A new identity involving Riemann-Liouville fractional integral is proposed. The result is then used to obtain some estimates of upper bound for a function associated with Riemann-Liouville fractional integral via h-convex functions. An application for establishing the inequalities related to special means is also considered.


Introduction
A set  ⊂ R is said to be convex, if A function  :  → R is said to be convex, if In 1978, Breckner [1] introduced the concept of -convex functions as a generalization of convex functions, as follows.
Convexity of a function plays a vital role in theory of inequalities, because many inequalities can easily be obtained using the functions having convexity properties.Hermite and Hadamard's result which is known as Hermite-Hadamard's inequality is one of the most fascinating results in the field of integral inequalities.This inequality provides a lower and an upper estimate for the integral average of any convex function defined on an interval.This famous result reads as follows.
Let  : [, ] → R be a convex function, then Sarikaya et al. [10] gave a generalization of Hermite-Hadamard's inequality using the ℎ-convexity of the function as follows.
Let  : [, ] → R be ℎ-convex function, then, for ℎ(1/2) ̸ = 0, we have Although the fractional calculus has a long history, it plays significant role in different fields of pure and applied mathematics [11].Up to now, the study of the fractional calculus is still very active.Sarikaya et al. [12] used the concepts of Riemann-Liouville integrals to obtain the fractional version of Hermite-Hadamard's inequality.In fact, there are numerous new inequalities that have been obtained using the techniques of fractional calculus.For more details, see [12][13][14][15].
In this paper, we present a new integral identity for differentiable functions involving fractional integrals.Then using this auxiliary result we establish our main results that are the estimates of upper bound for a function associated with Riemann-Liouville fractional integral via ℎ-convex functions.At the end of the paper, we give an application of the obtained results to the special means.
We begin with recalling the definition of Riemann-Liouville fractional integrals, as follows.

Main Results
In this section, we consider the estimates of upper bound for the function below, which is associated with Riemann-Liouville fractional integral.Consider the following: In order to establish the estimates of upper bound for Ψ(, ; ; )(), we first prove an auxiliary result which plays an important role in dealing with subsequent results. Proof.
Integrating  1 gives Similarly integrating  2 , one has Using ( 14) and ( 15) in ( 13) leads to the identity described in Lemma 4.
Based on Lemma 4, we are now in a position to establish our main results.
This completes the proof of Theorem 5.
We now discuss some special cases which can be deduced directly from Theorem 5.
(I) Putting ℎ() =  in Theorem 5, we have the following.] . ( The proof of Theorem 10 is complete.
We give now four corollaries that follow from the special cases of Theorem 10.

Application to Special Means
In this section, we give an application of the obtained results to special means.
(  (3) The extended logarithmic mean of two positive numbers ,  ( ̸ = ) is defined by We focus on the estimation of upper bound for the difference between logarithmic mean and arithmetic mean; we shall establish two inequalities related to these means. where Proof.We start by verifying that () =  −1 (0 <  ≤ 1) is -convex on (0, ∞).