A New Version of the Hermite–Hadamard Inequality for Riemann–Liouville Fractional Integrals

Pshtiwan Othman Mohammed,Iver Brevik

doi:10.3390/sym12040610

Abstract

Integral inequalities play a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods. Thus, the present days need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the field of inequalities due to the behavior of its definition. There is a strong relationship between convexity and symmetry. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the past few years. In this article, we firstly point out the known Hermite–Hadamard (HH) type inequalities which are related to our main findings. In view of these, we obtain a new inequality of Hermite–Hadamard type for Riemann–Liouville fractional integrals. In addition, we establish a few inequalities of Hermite–Hadamard type for the Riemann integrals and Riemann–Liouville fractional integrals. Finally, three examples are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.

Highlights

IntroductionIntegral inequalities form a strong and thriving field of study within the huge field of mathematical analysis
Integral inequalities form a strong and thriving field of study within the huge field of mathematical analysis. They have participated in the study of many common fields, e.g., ordinary differential equations, integral equations, and partial differential equations [1,2]. They have participated in the field of fractional differential equations, especially fractional integral inequalities, which have been crucial in providing bounds to solve initial and boundary value problems in fractional calculus, and in establishing the existence and uniqueness of solutions for certain fractional differential equations [3,4,5,6]
We present the Riemann–Liouville (RL) definition to facilitate the discussion of the aforementioned operations, which is most commonly used for fractional derivatives and integrals

Summary

Introduction

Integral inequalities form a strong and thriving field of study within the huge field of mathematical analysis. In 2013, Sarikaya et al [12] generalized the Hermite–Hadamard inequality in Equation (2) to fractional integrals of Riemann–Liouville type. Their result is stated as follows: o. Sarikaya and Yildirim [13] introduced a new version of the HH-inequality (Equation (2)) for Riemann–Liouville fractional integrals: 2η −1 Γ ( η + 1 ) They obtained some inequalities of midpoint type in the same paper. There are many papers studying integral inequalities for the Riemann–Liouville fractional integrals and some new relevant generalizations of Hermite–Hadamard type inequalities (see [11,12,19,21,22,23,24,25,26,27] for more details). The purpose of this paper is to introduce a new inequality of Hermite–Hadamard type and to establish a few related inequalities for Riemann–Liouville fractional integrals

The New Hermite–Hadamard Inequality

Examples

Conclusions