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

McSylvester Ejighikeme Omaba,Eze R. Nwaeze

doi:10.2478/amsil-2020-0026

Abstract

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.

Highlights

IntroductionKey words and phrases: Hermite–Hadamard inequalities, generalized Katugampola fractional integrals, generalized Riemann–Liouville fractional integral, convex and positive stochastic processes
We provide new Hermite–Hadamard type estimates for expectations of integrals of convex and p-convex stochastic processes
We start by presenting a generalization of Theorem 1.7

Summary

Introduction

Key words and phrases: Hermite–Hadamard inequalities, generalized Katugampola fractional integrals, generalized Riemann–Liouville fractional integral, convex and positive stochastic processes

Objectives

Results

Conclusion