Generalizations of some fractional integral inequalities via generalized Mittag-Leffler function

Fractional inequalities are useful in establishing the uniqueness of solution for partial differential equations of fractional order. Also they provide upper and lower bounds for solutions of fractional boundary value problems. In this paper we obtain some general integral inequalities containing generalized Mittag-Leffler function and some already known integral inequalities have been produced as special cases.


Introduction
Inequalities play a vital role in both pure and applied mathematics. Specially, inequalities involving the derivative and the integral of functions are very captivating for researchers. Convex functions play an important role in the study of inequalities in all kinds of mathematical analysis.
Definition  A function f : I → R, where I is an interval in R, is said to be a convex function if holds for t ∈ [, ] and x, y ∈ I. Theorem . Let a function f : I → R be convex on I. Then we have where a, b ∈ I, a < b.
In the literature this inequality is known as the Hadamard inequality.
Recently, a number of researchers have taken great interest in establishing the Hadamard type inequalities for fractional integral operators of different kinds in the diverse field of fractional calculus. For example one may refer to [-].

Fractional derivative and integral operators
Fractional calculus is a theory of integral and differential operators of non-integral order. Many mathematicians, like Liouville, Riemann and Weyl, made major contributions to the theory of fractional calculus. The study on the fractional calculus continued with the contributions from Fourier, Abel, Lacroix, Leibniz, Grunwald and Letnikov. For details, see [, , ]. A first formulation of an integral operator of fractional order in reliable form is named the Riemann-Liouville fractional integral operator.
Then Riemann-Liouville k-fractional integrals of f of order ν >  with a ≥  are defined by and Actually, these forms of fractional integral operators have been formulated due to the work of Sonin [], Letnikov [] and then by Laurent []. Now a days a variety of fractional integral operators are under discussion. Many generalized fractional integral oper-ators also take part in generalizing the theory of fractional integral operators [, , , , , -].
Definition  ([]) Let μ, ν, k, l, γ be positive real numbers and ω ∈ R. Then the generalized fractional integral operator containing the generalized Mittag-Leffler function γ ,δ,k μ,ν,l,ω,a + and γ ,δ,k μ,ν,l,ω,b -for a real valued continuous function f is defined by where the function E γ ,δ,k μ,ν,l is the generalized Mittag-Leffler function defined as We use this definition of S in the sequel in our results. A lot of authors presently are working on inequalities involving fractional integral operators, for example the versions of Riemann-Liouville, Caputo, Hillfer, Canvati etc. In fact fractional integral inequalities are useful in establishing the uniqueness of solutions for partial differential equations of fractional order, also they provide upper and lower bounds for solutions of fractional boundary value problems.
In this paper we give some integral inequalities for a generalized fractional integral operator containing the generalized Mittag-Leffler function which are generalizations of several results proved in [-].
The following result was obtained by Sarikaya et al. in [].
, then the following inequalities for a fractional integral hold:

Main results
First of all we establish the following result which would be helpful to obtain the main result.
Since |f | is convex function, it can be written as After simplification of inequality () we get the result.
Proof By using Lemma ., we have Using the Hölder inequality, we have b a g(s)E γ ,δ,k μ,ν,l ωs μ ds Using g ∞ = sup t∈ [a,b] |g(t)| and absolute convergence of the generalized Mittag-Leffler Since |f (t)| q is convex function, we have After simplification, we get the required result.
Remark . For particular values of the parameters, Theorem . gives the following results. where ω =  μ ω (b-a) μ .
Proof Since f is a convex function, we have Multiplying both sides of () by t ν- E