Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals

Some Hermite-Hadamard type inequalities for generalized k-fractional integrals (which are also named (k,s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(k,s)$\end{document}-Riemann-Liouville fractional integrals) are obtained for a fractional integral, and an important identity is established. Also, by using the obtained identity, we get a Hermite-Hadamard type inequality.


Introduction
Let f : I ⊆ R → R be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. The following inequality with α > , where the symbols J α a + and J α b -denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order α ∈ R + that are defined by respectively. Here (·) denotes the classical gamma function [], Chapter .
, then the following inequality for Riemann-Liouville fractional integrals holds: The Pochhammer k-symbol (x) n,k and the k-gamma function k are defined as follows (see []): where kZ - := {kn : n ∈ Z - }. It is noted that the case k =  of (.) and (.) reduces to the familiar Pochhammer symbol (x) n and the gamma function . The function k is given by the following integral: The function k defined on R + is characterized by the following three properties: We want to recall the preliminaries and notations of some well-known fractional integral operators that will be used to obtain some remarks and corollaries.
The most important feature of (k, s)-fractional integrals is that they generalize some types of fractional integrals (Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral, generalized fractional integral and Hadamard fractional integral). These important special cases of the integral operator s k J α a are mentioned below. () For k = , the operator in (.) yields the following generalized fractional integrals defined by Katugompola in []: Firstly by taking k = , after that by taking limit r → - + and using L'Hôpital's rule, the operator in (.) leads to the Hadamard fractional integral operator [, ]. That is, This relation is as follows: (.) () Again, taking s =  and k = , operator (.) gives us the Riemann-Liouville fractional integration operator In recent years, these fractional operators have been studied and used to extend especially Grüss, Chebychev-Grüss and Pólya-Szegö type inequalities. For more details, one may refer to the recent works and books [, -].

Main results
Let f : I • → R be a given function, where a, b ∈ I • and  < a < b < ∞. We suppose that f ∈ L ∞ (a, b) such that s k J α a + f (x) and s k J α b -f (x) are well defined. We define functions Hermite-Hadamard's inequality for convex functions can be represented in a (k, s)fractional integral form as follows by using the change of variables u = t-a x-a ; we have from Proof For u ∈ [, ], let ξ = au + (u)b and η = (u)a + bu. Using the convexity of f , we get That is, Now, multiplying both sides of (.) by and integrating over (, ) with respect to u, we get Using the identitỹ and from (.), we obtain Accordingly, we have Similarly, multiplying both sides of (.) by integrating over (, ) with respect to u, and from (.), we also get By adding inequalities (.) and (.), we get which is the left-hand side of inequality (.).
Since f is convex, for u ∈ [, ], we have Multiplying both sides of (.) by and integrating over (, ) with respect to u, we get That is, Similarly, multiplying both sides of (.) by and integrating over (, ) with respect to u, we also get Adding inequalities (.) and (.), we obtain which is the right-hand side of inequality (.). So the proof is complete.
We want to give the following function that we will use later: For α, k >  and s ∈ R \{-}, let ∇ α,s : [, ] → R be the function defined by In order to prove our main result, we need the following identity.
Lemma . Let α, k >  and s ∈ RI • . If f is a differentiable function on I • such that f ∈ L[a, b] with a < b, then we have the following identity: Proof Using integration by parts, we obtain Similarly, we get Using the fact that F(x) = f (x) +f (x) and by simple computation, from equalities (.) and (.), we get Note that we have