On Extended General Mittag–Leffler Functions and Certain Inequalities

Marcela V Mihai,Tingsong Du,Muhammad Uzair Awan,Artion Kashuri,Muhammad Aslam Noor,Khalida Inayat Noor

doi:10.3390/fractalfract3020032

Abstract

In this paper, we introduce and investigate generalized fractional integral operators containing the new generalized Mittag–Leffler function of two variables. We establish several new refinements of Hermite–Hadamard-like inequalities via co-ordinated convex functions.

Highlights

Introduction and Preliminaries The HermiteHadamard inequality states that if a function Ψ : I ⊆ R → R is convex, Ψ a+b 2 ≤1 b−a b Ψ(x) dx ≤Ψ(a) + Ψ(b), a (1)where a, b ∈ I with a < b
Many researchers have turned their attention to the Hermite–Hadamard inequality and have found many variations and generalizations of it via various types of convexity
A function Ψ : ∆ → R will be called convex on the coordinates if the partial mappings Ψy : [a, b] → R, Ψy(u) = Ψ(u, y) and Ψx : [c, d] → R, Ψx(v) = Ψ(x, v) are convex where defined for all y ∈ [c, d], and x ∈ [a, b]

Summary

We also make the convention

Salim and Faraj [7] have defined the generalized fractional integral operators containing Mittag–Leffler functions: Definition 3. The generalized fractional integral operators containing Mittag–Leffler function εγμ,,δν,,kl,ω,a+ and εγμ,,δν,,kl,ω,b− for a real-valued continuous function Ψ are defined by: εγμ,,δν,,kl,ω,a+ Ψ (x) =. Respectively, where the function Eμγ,,νδ,,lk is a generalized Mittag–Leffler function defined as. If k = l = 1 in (3), the integral operator εγμ,,δν,,k1,ω,a+ Ψ reduces to an integral operator εγμ,,δν,,1l,ω,a+ Ψ containing generalized Mittag–Leffler function Eμγ,,νδ,, introduced by Srivastava and Tomovski in [8]. Let μ, ν, k, l, γ be positive real numbers and ω ∈ R, εγμ11,,δν11,,kl11,ω1,a+.

If the function tG

If tG

Then we have inequality

Conclusions