A new refinement of the arithmetic mean -- geometric mean inequality
type: Item
instance of: scholarly article
title: A new refinement of the arithmetic mean -- geometric mean inequality
zbMATH Open document ID: 0897.26004
author: Horst Alzer
published in: Rocky Mountain Journal of Mathematics
publication date: 1998-05-25T00:00:00Z
full work available at URL: CONT27-3.html
review text: The following inequality is offered as main result. Let \(x_{j}\) be positive reals, \(p_{j}\) \((j=1,\dots,n)\) nonnegative reals with sum 1, \(A\) the arithmetic, \(G\) the geometric mean of the \(x_{j}\) with weight \(p_{j}\) and \(M\) the maximum of the \(x_{j}\) \((j=1,\dots,n)\). Then \[ \sum_{j=1}^{n} p_{j}(x_{j}-G)^{2}\leq 2M(A-G) \] with equality iff all \(x_j\) with positive coefficients are equal.
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1143462
zbMATH Keywords: inequalities
zbMATH Keywords: arithmetic mean
zbMATH Keywords: maximum
zbMATH Keywords: geometric mean
zbMATH Keywords: weight
reviewed by: János Aczél
MaRDI profile type: Publication
OpenAlex ID: W2071243106
cites work: A Refinement of the Arithmetic Mean-Geometric Mean Inequality
cites work: An Elementary Proof of the Square Summability of the Discrete Hilbert Transform
DOI: 10.1216/RMJM/1181071887
description: scientific article; zbMATH DE number 1143462