Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

Pshtiwan Othman Mohammed,Artion Kashuri,Shengda Zeng,Thabet Abdeljawad

doi:10.3390/sym12091485

Abstract

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of φ-Riemann–Liouville fractional integral operators and the recently defined class of convex functions, namely the σ˘-convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a σ˘-convex function with respect to an increasing function involving the φ-Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out.

Highlights

First of all, we recall the basic notations in convex analysis
After introducing Hermite–Hadamard type inequalities (2) and (3), many classical and fractional integral inequalities have been established by a huge number of researcher; for more details one can see References [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]
We have considered a new class of convex functions and the definition of φ−RL

Summary

Introduction

We recall the basic notations in convex analysis. A set χ ⊂ R is said to be convex if η u + (1 − η )v ∈ χ, for each u, v ∈ χ and η ∈ [0, 1]. After introducing Hermite–Hadamard type inequalities (2) and (3), many classical and fractional integral inequalities have been established by a huge number of researcher; for more details one can see References [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. The aim of this article is to establish several inequalities of Hermite–Hadamard’s type for σ-convex functions via φ-Riemann–Liouville (φ−RL) fractional integrals, where the φ−RL fractional integrals are defined as follows (see e.g., References [31,32]).

Hermite–Hadamard’s Type Inequalities for σ-Convex Functions

Further Consequences

The Modified Bessel Functions

Special Means

Conclusions