New Estimations of Hermite–Hadamard Type Integral Inequalities for Special Functions

Hijaz Ahmad,Muhammad Tariq,Clemente Cesarano,Soubhagya Kumar Sahoo,Jamel Baili

doi:10.3390/fractalfract5040144

Abstract

In this paper, we propose some generalized integral inequalities of the Raina type depicting the Mittag–Leffler function. We introduce and explore the idea of generalized s-type convex function of Raina type. Based on this, we discuss its algebraic properties and establish the novel version of Hermite–Hadamard inequality. Furthermore, to improve our results, we explore two new equalities, and employing these we present some refinements of the Hermite–Hadamard-type inequality. A few remarkable cases are discussed, which can be seen as valuable applications. Applications of some of our presented results to special means are given as well. An endeavor is made to introduce an almost thorough rundown of references concerning the Mittag–Leffler functions and the Raina functions to make the readers acquainted with the current pattern of emerging research in various fields including Mittag–Leffler and Raina type functions. Results established in this paper can be viewed as a significant improvement of previously known results.

Highlights

The Hermite–Hadamard inequality, which is the primary consequence of convex functions having a beautiful geometrical understanding and broad use, has stood out with incredible interest in fundamental mathematics
Numerous mathematicians have given their endeavors to normalization, refining, impersonation, and extension of the Hermite–Hadamard inequality using different types of novel convexities
We addressed a novel idea for the generalized preinvex function, namely the s-type preinvex function

Summary

Introduction

The Hermite–Hadamard inequality, which is the primary consequence of convex functions having a beautiful geometrical understanding and broad use, has stood out with incredible interest in fundamental mathematics. Integral inequalities on the Raina function have been a subject of discussion for a significant length of time Because of their possibilities to be extended, a few variations have been set up by many mathematicians, see references [6,7]. The theory of convex mappings has a wide scope of possible applications in many interesting and captivating fields of exploration This theory likewise assumes an eminent part in different areas, such as information theory, coding theory, engineering, Fractal Fract. Guessab et al [8,9,10] worked on the error estimations and multivariate approximation theory This hypothesis has an amazing commitment to the expansions and enhancements of various areas of numerical and applied sciences. For some of the recent considerations, we refer to the references [11,12,13,14]

Preliminaries

Generalized s–Type Convex Function of Raina Type and Its Properties

Hermite–Hadamard Type Inequality via Generalized s–Type Convex Function of

Refinements of Hermite–Hadamard Type Inequality

Conclusions