On new general versions of Hermite\u2013Hadamard type integral inequalities via fractional integral operators with Mittag-Leffler kernel

Havva Kavurmacı Önalan,Merve Avcı Ardıç,Dumitru Baleanu,Ahmet Ocak Akdemir

doi:10.1186/s13660-021-02721-9

Abstract

The main motivation of this study is to bring together the field of inequalities with fractional integral operators, which are the focus of attention among fractional integral operators with their features and frequency of use. For this purpose, after introducing some basic concepts, a new variant of Hermite–Hadamard (HH-) inequality is obtained for s-convex functions in the second sense. Then, an integral equation, which is important for the main findings, is proved. With the help of this integral equation that includes fractional integral operators with Mittag-Leffler kernel, many HH-type integral inequalities are derived for the functions whose absolute values of the second derivatives are s-convex and s-concave. Some classical inequalities and hypothesis conditions, such as Hölder’s inequality and Young’s inequality, are taken into account in the proof of the findings.

Highlights

Mathematics has basically started its adventure as a theoretical field with the efforts of researchers for centuries, and has continuously aimed to formulate events and phenomena in various fields such as physics, engineering, modeling, and mathematical biology into a form that can be calculated
Definition 2 The fractional integral related to the new fractional derivative with nonlocal kernel of a mapping θ ∈ H1(φ1, φ2) is defined as follows: AB φ1
After giving some basic information and concepts about fractional analysis, which is one of the basic foundations of the study, we will continue by reminding some basic concepts on convex functions and inequalities

Summary

Introduction

Mathematics has basically started its adventure as a theoretical field with the efforts of researchers for centuries, and has continuously aimed to formulate events and phenomena in various fields such as physics, engineering, modeling, and mathematical biology into a form that can be calculated. Definition 2 (see [5]) The fractional integral related to the new fractional derivative with nonlocal kernel of a mapping θ ∈ H1(φ1, φ2) is defined as follows: AB φ1 After giving some basic information and concepts about fractional analysis, which is one of the basic foundations of the study, we will continue by reminding some basic concepts on convex functions and inequalities.

Results

Conclusion