Notes on the Schur-convexity of the extended mean values
type: Item
instance of: scholarly article
title: Notes on the Schur-convexity of the extended mean values
zbMATH Open document ID: 1086.26008
author: Feng Qi
author: Anthony Sofo
author: József Sándor
author: Silvestru Sever Dragomir
published in: Taiwanese Journal of Mathematics
publication date: 2006-01-26T00:00:00Z
review text: Let \(f\) be a continuous function on an interval \(I\) and \(p\) be a continuous weight on \(I\).The weighted arithmetic mean of \(f\) is defined by \[ F(x,y)=\begin{cases} \frac{\int_{x}^{y}p(t)f(t)dt}{\int_{x}^{y}p(t)dt},&x\neq y \\ f(x),&x=y. \end{cases} \] The authors prove that \(F\) is Schur-convex (Schur-concave) on \(I^{2}\) if and only if the inequality \[ F(x,y)\leq \frac{p(x)f(x)+p(y)f(y)}{p(x)+p(y)} \] holds (reverses) for all \(x,y\in I.\) Schur-convexity of the extended mean values is also studied.
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 26D07
Mathematics Subject Classification ID: 26D20
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 5001901
zbMATH Keywords: extended mean values
zbMATH Keywords: Schur convexity
zbMATH Keywords: weighted arithmetic mean of a function
reviewed by: Gheorghe Toader
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500407849
OpenAlex ID: W2150601196
DOI: 10.11650/TWJM/1500407849
description: scientific article; zbMATH DE number 5001901