Integral inequalities in fractional calculus with general analytic kernels

Abstract

Many different definitions of fractional calculus have been proposed in the literature, especially in recent years, and these can be classified into groups with similar properties. One direction of research has involved proving inequalities for fractional integrals of particular types of functions, such as Hermite--Hadamard inequalities and related results. Here we provide theorems valid for a whole general class of fractional operators (anything defined using an integral with an analytic kernel function), so that it is no longer necessary to prove such results for each model one by one. We consider several types of fractional integral inequalities, which apply to functions of convex and synchronous type, and extend them to the full generality of fractional calculus with analytic kernels.