On Hadamard Type Fractional Inequalities for Riemann–Liouville Integrals via a Generalized Convexity

Tao Yan,Hafsa Yasmeen,Ghulam Farid,Chahn Yong Jung

doi:10.3390/fractalfract6010028

Abstract

In the literature of mathematical inequalities, convex functions of different kinds are used for the extension of classical Hadamard inequality. Fractional integral versions of the Hadamard inequality are also studied extensively by applying Riemann–Liouville fractional integrals. In this article, we define (α,h−m)-convex function with respect to a strictly monotone function that unifies several types of convexities defined in recent past. We establish fractional integral inequalities for this generalized convexity via Riemann–Liouville fractional integrals. The outcomes of this work contain compact formulas for fractional integral inequalities which generate results for different kinds of convex functions.

Highlights

Riemann–Liouville Integrals via a Convexity theory has a rich history and has emerged as a powerful tool over the last century
We define a new class of functions, named as (α, h − m)−convex function with respect to a strictly monotone function. This class of functions will unify several types of convexities, the Hadamard fractional integral inequalities are established for this new class of functions keeping Riemann–Liouville fractional integrals
The classical Riemann–Liouville integrals of fractional order and the Hadamard inequality for these integrals are given in the following definition and theorems, respectively: Definition 2

Summary

Introduction

Riemann–Liouville Integrals via a Convexity theory has a rich history and has emerged as a powerful tool over the last century. We define a new class of functions, named as (α, h − m)−convex function with respect to a strictly monotone function This class of functions will unify several types of convexities, the Hadamard fractional integral inequalities are established for this new class of functions keeping Riemann–Liouville fractional integrals. The classical Riemann–Liouville integrals of fractional order and the Hadamard inequality for these integrals are given in the following definition and theorems, respectively: Definition 2. The left and right sided k-fractional Riemann–Liouville integrals of the function f of fractional order μ > 0, k > 0 are defined as follows:. We have deduced a lot of fractional versions of Hadamard inequalities published in [1,4,7,8,9,10,11,14,15,27,28,29,30,31,32]

Some New Notions of Convexity

Conclusions

Methods