Verschärfung einer Ungleichung von Ky Fan
type: Item
instance of: scholarly article
title: Verschärfung einer Ungleichung von Ky Fan
zbMATH Open document ID: 0655.26011
DOI: 10.1007/BF01836094
author: Horst Alzer
published in: Aequationes Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137287
review text: In this paper we prove the following: If \(A_ n\), \(G_ n\) and \(H_ n\) (resp. \(A_ n'\), \(G_ n'\) and \(H_ n')\) denote the arithmetic, geometric and harmonic means of \(a_ 1,...,a_ n\) (resp. \(1-a_ 1,...,1-a_ n\)) and if \(a_ i\in (0,1/2]\), \(i=1,...,n,\) then \[ (*)\quad (G_ n/G_ n')^ n\leq (A_ n/A_ n')^{n-1}H_ n/H_ n', \] with equality holding for \(n=1,2\). For \(n\geq 3\) equality holds if and only if \(a_ 1=...=a_ n\). The inequality (*) sharpens the well-known inequality of Ky Fan: \(G_ n/G_ n'\leq A_ n/A_ n'\).
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 4069233
zbMATH Keywords: arithmetic means
zbMATH Keywords: geometric means
zbMATH Keywords: harmonic means
zbMATH Keywords: inequalities
zbMATH Keywords: inequality of Ky Fan
Wikidata QID: Q55983127
MaRDI profile type: Publication
cites work: ÜBER DIE UNGLEICHUNG ZWISCHEN DEM GEOMETRISCHEN UND DEM ARITHMETISCHEN MITTEL
cites work: On an inequality of Ky Fan
cites work: A class of means and related inequalities
cites work: Q3213358
cites work: Generalization of an inequality of Ky Fan
cites work: On a Ky Fan inequality of the complementary A-G type and its variants
cites work: Functional equation approach to inequalities. II
OpenAlex ID: W2080704849
description: scientific article; zbMATH DE number 4069233