Inequalities for sums of multipowers
type: Item
instance of: scholarly article
title: Inequalities for sums of multipowers
zbMATH Open document ID: 0723.26010
DOI: 10.1007/BF01903719
author: Zsolt Páles
published in: Acta Mathematica Hungarica
publication date: 1990-01-01T00:00:00Z
review text: The author offers generalizations of Hölder's inequality in the form \[ \prod^{m}_{j=0}(\sum^{n}_{i=1}x_{(i)}^{a_{(j)}}/n)^{\gamma _ j}\geq 1, \] where \(\gamma_ j\in {\mathbb{R}}\); \(a_{(j)}\in {\mathbb{R}}^ k\), \(x_{(i)}\in {\mathbb{R}}^ k_+\); \(i=1,...,n\); \(j=0,1,...,m\), and \[ x^ a=(x_ 1,...,x_ k)^{(a_ 1,...,a_ k)}:=\prod^{k}_{\ell =1}x_{\ell}^{a_{\ell}}. \] Note: \(x_ n\) in the line 4 of p. 165 should be \(x_ k\), \({\mathbb{R}}_+\) at the end of line 12 of the same page should be \({\mathbb{R}}^ k_+\). More importantly, the condition, under which the above inequality holds, follows considerably later, in the Corollary on p. 168, later quoted as ``Corollary 1'': \(\sum^{m}_{j=0}\gamma_ jf(a_{(j)})\geq 0\) for all convex f with \(f(0)=0\). If \(\sum^{m}_{j=0}\gamma_ j=0\) and \(k=1\) then this strong condition is weakened to \(\sum^{n}_{j=0}\gamma_ j| a_ j-a_ r| \geq 0\quad (r=0,...,n).\) Several known inequalities are sketchily deduced from this result. The nice idea would deserve a more careful exposition.
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26A51
zbMATH DE Number: 4192051
zbMATH Keywords: means
zbMATH Keywords: sums
zbMATH Keywords: products
zbMATH Keywords: powers
zbMATH Keywords: convex functions
zbMATH Keywords: generalizations of Hölder's inequality
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01903719
OpenAlex ID: W2002783802
description: scientific article; zbMATH DE number 4192051