Hermite-Hadamard type inequalities for \(n\)-polynomial generalized convex functions of Raina type and some related inequalities

Saad Ihsan Butt,Muhammad Tariq,Muhammad Nadeem

doi:10.30538/psrp-easl2021.0074

Abstract

In this paper, we introduce the concept of a new family of convex functions namely n-polynomial generalized convex functions of Raina type. We investigate the algebraic properties of a newly introduced idea and discuss their connections with convex functions. Furthermore, we establish the new version of Hermite–Hadamard and some refinements of Hermite-Hadamard type inequalities this class of functions. Finally, we investigate some applications to special means of real numbers. Results obtained in this paper can be viewed as a significant improvement of previously known results and also may stimulate and energize for further activities in this research area field.

Highlights

The term “convexity" is the most important, interesting, natural, and fundamental notations in mathematics and was used for the first time widely in the classical book by Hardy, Little, and Polya
The aim of this paper is to introduce a new family of convex functions namely n–polynomial generalized convex functions of Raina type
We have introduced a new family of convex functions namely n–poly generalized convex functions of Raina type

Summary

Introduction

The term “convexity" is the most important, interesting, natural, and fundamental notations in mathematics and was used for the first time widely in the classical book by Hardy, Little, and Polya (see [1]). Many researchers always try to do and use new ideas for the enjoyment and beautification of convex analysis. This theory provides us with interesting and powerful numerical tools, on the basis of these tools, we solve a lot of problems that appear in mathematics. Due to its diverse and widespread applications, the ideas about the topics of convexity and integral inequalities have been extended, improved, and generalized in many different ways and the researchers are always inspired by the relationship of these two fields and many new inequalities have been obtained via the convexity property of the functions. Interesting algebraic properties, several new integral inequalities, example with logic, and applications to means via the newly introduced class of functions are provided

Preliminaries

New inequalities for n–polynomial generalized convex functions of Raina type

The harmonic mean

Conclusion