Majorization versus power majorization
type: Item
instance of: scholarly article
title: Majorization versus power majorization
zbMATH Open document ID: 0698.26010
DOI: 10.1007/BF01909366
published in: Analysis Mathematica
publication date: 1986-01-01T00:00:00Z
review text: Let \({\mathfrak x}=(x_ 1,...,x_ n)\) and \({\mathfrak y}=(y_ 1,...,y_ n)\) be n-tuples of positive real numbers and recall that \({\mathfrak x}\) is said to be majorized by \({\mathfrak y}\) provided that \((1)\quad \sum^{n}_{i=1}\phi (x_ i)\leq \sum^{n}_{i=1}\phi (y_ i)\) for every convex function \(\phi\) : (0,\(\infty)\to {\mathbb{R}}\). Several useful characterizations of majorization have been given and much of the general theory of inequalities may be derived from these ideas. On the other hand, in view of the central role of \(\ell^ p\)-means in this theory, the following weaker concept is also appealing. We say that \({\mathfrak x}\) is power majorized by \({\mathfrak y}\) provided that (2) \(\sum^{n}_{i=1}x^ p_ i\leq \sum^{n}_{i=1}y^ p_ i\), whenever \(p\in {\mathbb{R}}\) (with reversal of the inequality sign when \(0<p<1).\) It can be shown that (1) and (2) are equivalent for \(n\leq 3\), and it is natural to ask whether this equivalence holds for all positive integers n. The purpose of this note is to give a counterexample. Our construction is based on a non-homogeneous version of a familiar \(\ell^ p\)-space inequality.
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 4144306
zbMATH Keywords: power majorization
zbMATH Keywords: inequalities
zbMATH Keywords: \(\ell ^ p\)-space inequality
author: Grahame Bennett
MaRDI profile type: Publication
cites work: Inequalities: theory of majorization and its applications
full work available at URL: https://doi.org/10.1007/bf01909366
OpenAlex ID: W1862650117
description: scientific article; zbMATH DE number 4144306