Refinements of Majorization Inequality Involving Convex Functions via Taylor’s Theorem with Mean Value form of the Remainder

Shanhe Wu,Muhammad Adil Khan,Hidayat Ullah Haleemzai

doi:10.3390/math7080663

Shanhe Wu, Muhammad Adil Khan + Show 1 more

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https://doi.org/10.3390/math7080663

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Journal: Mathematics	Publication Date: Jul 24, 2019
Citations: 8	License type: CC BY 4.0

Affiliation: Longyan University, University of Peshawar

Abstract

The aim of this paper is to establish some refined versions of majorization inequality involving twice differentiable convex functions by using Taylor theorem with mean-value form of the remainder. Our results improve several results obtained in earlier literatures. As an application, the result is used for deriving a new fractional inequality.

Highlights

The notion of majorization was introduced in the celebrated monograph [1] by Hardy, Littlewood and Pólya, which was used as a measure of the diversity of the components of an n-dimensional vector
In this paper we focus on a type of majorization inequality involving convex functions, which reveals the correlations among majorization, convex functions and inequalities
The following classical majorization inequality can be found in the monographs of Marshall and

Summary

Introduction

The authors have given considerable attention to the generalizations and applications of the majorization and related inequalities, for details, we refer the reader to our papers [3,4,5,6,7,8,9,10,11,12,13]. The following classical majorization inequality can be found in the monographs of Marshall and Fuchs [15] gave a weighted generalization of the majorization theorem, as follows: Mathematics 2019, 7, 663; doi:10.3390/math7080663 If Ψ : I → R is a continuous increasing convex function, n n i =1 i =1

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