Generalized Fractional Inequalities of the Hermite–Hadamard Type for Convex Stochastic Processes

Abstract

A generalization of the Hermite–Hadamard (HH) inequality for a positive convex stochastic process, by means of a newly proposed fractional integral operator, is hereby established. Results involving the Riemann– Liouville, Hadamard, Erdélyi–Kober, Katugampola, Weyl and Liouville fractional integrals are deduced as particular cases of our main result. In addition, we also apply some known HH results to obtain some estimates for the expectations of integrals of convex and p-convex stochastic processes. As a side note, we also pointed out a mistake in the main result of the paper [Hermite–Hadamard type inequalities, convex stochastic processes and Katugampola fractional integral, Revista Integración, temas de matemáticas 36 (2018), no. 2, 133–149]. We anticipate that the idea employed herein will inspire further research in this direction.

NEW ψ - & (k, r)- FRACTIONAL CONFORMABLE INTEGRALS AND INEQUALITIES OF THE HERMITE–HADAMARD TYPE FOR POSITIVE–CONVEX STOCHASTIC PROCESSES

“…Sarikaya et al in [15] introduced the (k, r)-fractional integral of the Riemann-type (see also [2,6,10,16]); and the H-H inequality for the (k, r)-fractional integral of the Riemann-type for a convex stochastic process was recently given by [13] as follows:…”

“…One may also refer to the same paper [13] where the authors in a recent result, generalized Theorem 2.3 for the Katugampola generalized fractional integrals defined in [17,18], as follows:…”

“…Hermite-Hadamard (H-H) integral inequality is one of the most famous inequalities for convex functions in the theory of fractional integral inequalities. There have been various versions and generalizations for the H-H inequality for convex functions, and recently an extension to convex stochastic processes for different fractional integrals, (see [3,5,8,11,12,13,14] and their references). In [6], the authors considered the H-H type inequalities for generalized k-fractional conformable integrals for convex functions.…”