Bounds of Riemann-Liouville Fractional Integrals in General Form via Convex Functions and Their Applications

Ghulam Farid,Shin Kang,Muhammad Saleem,Waqas Nazeer,Sajid Mehmood

doi:10.3390/math6110248

Abstract

In this article, we establish bounds of sum of the left and right sided Riemann Liouville (RL) fractional integrals and related inequalities in general form. A new and novel approach is followed to obtain these results for general Riemann Liouville (RL) fractional integrals. Monotonicity and convexity of functions are used with some usual and straight forward inequalities. The presented results are also have connection with some known and already published results. Applications and motivations of presented results are briefly discussed.

Highlights

The aim of this paper is to study several fractional integral operators in Fractional Calculus via convexity
A days a variety of fractional integral operators are under discussion and many generalized fractional integral operators take a part in generalizing the theory of fractional calculus [4,5,6,7,8]
Further objective of this paper is to find bounds of the sum of the left-sided and right-sided (RL) fractional integrals and related inequalities in general form which provide in particular results for fractional integrals defined by Katugampola et al in [11], Jarad et al in [12] and

Summary

Introduction

The aim of this paper is to study several fractional integral operators in Fractional Calculus via convexity. Our focus is to use convexity property of functions as well as of absolute values of their derivatives in the establishment of bounds of Riemann-Liouville fractional integrals in general form defined in Definition 1. First we give the following fractional integral inequality which provides the bound of sum of the left and the right sided (RL) fractional integrals in general form. To obtain this result we use the convexity and monotonicity of used functions.

Applications

Conclusions