Study of fractional integral inequalities involving Mittag-Leffler functions via convexity

Zhihua Chen,Maryam Saddiqa,Naveed Latif,Saleem Ullah,Ghulam Farid

doi:10.1186/s13660-020-02465-y

Abstract

This paper studies fractional integral inequalities for fractional integral operators containing extended Mittag-Leffler (ML) functions. These inequalities provide upper bounds of left- and right-sided fractional integrals for (alpha, h-m) convex functions. A generalized fractional Hadamard inequality is established. All the results hold for h-convex, (h, m)-convex, (alpha, m)-convex, (s, m)-convex, and associated functions.

Highlights

Convexity was introduced at the beginning of the twentieth century
Due to having many fascinating and important properties, a convex function plays a vital role in almost all areas of mathematical analysis, probability theory, optimization theory, graph theory, etc
It has been defined in different convenient ways, for example, graph of a convex function always lies below the chord joining any two points lying on its graph, the derivative of a differentiable convex function is increasing and vice versa, a convex function has line of support at each point of the interior of its domain, and many others

Summary

Introduction

Due to having many fascinating and important properties, a convex function plays a vital role in almost all areas of mathematical analysis, probability theory, optimization theory, graph theory, etc. A function f : I ⊆ R → R satisfying the inequality f (ta + (1 – t)b) ≤ tf (a) + (1 – t)f (b), where I is an interval, t ∈ [0, 1], and a, b ∈ I, is called convex This analytic form of presentation of a convex function motivated the authors to define other types of convex functions for example m-convex, s-convex, (s, m)-convex, h-convex, (h, m)-convex, (α, m)-convex, exponentially convex, etc. In this age convex functions lead to the theory of convex analysis, theory of inequalities, a lot of research articles and books are dedicated to the literature which has been developed due to convex function, see [1, 3, 4, 20, 22, 25, 31]

Objectives

Conclusion