New characterizations of some mean-values
type: Item
instance of: scholarly article
title: New characterizations of some mean-values
zbMATH Open document ID: 0878.39005
author: Themistocles M. Rassias
author: Hiroshi Haruki
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-01-08T00:00:00Z
review text: \textit{H. Haruki} proved in Publ. Math. 38, No. 3/4, 323-332 (1991; Zbl 0746.39004) among others that \(p^{-1} (\int_0^{2\pi} p(s)dt/(2\pi))\) (\(p\) is a real valued \(C^2\) function on the positive real and \(s=(a^2\cos^2t+ b^2\sin^2t)^{1/2})\) equals the arithmetic or geometric mean or the root-mean-square of \(a,b\) (no weights) if and only if \(p(s)= A\log s+B\), \(=As^{-2}+B\) or \(=As^2+B\), respectively. In the present paper the authors first choose \(s=a\sin^2t+b\cos^2t\) and offer that in this case the above integral mean equals the arithmetic or geometric mean or the square-mean-root or the geometric mean of the harmonic and of the geometric mean of \(a,b\) iff \(q(s)= As+B\), \(=As^{-1}+B\), \(=A\log s+B\) or \(=As^{-2}+B\), respectively. Similar results are offered for \(s=(a^{-1} \sin^2t+b^{-1}\cos^2t)^{-1}\). Reviewer's remark: In Theorem 2(iii) + should stand in place of \(-\). Also, in the introduction the authors state ``postulates'' for mean values \(M\) of two variables \(a,b\). One of the postulates requires that \(M\) is defined for all positive \(a,b\), the second that \(M\) be a symmetric function, the third that \(M(a,a)=a\). Surprisingly, it is not postulated that the value \(M(a,b)\) lie between those of \(a\) and \(b\) (strictly or otherwise).
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 45A05
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 930256
zbMATH Keywords: arithmetic mean
zbMATH Keywords: harmonic mean
zbMATH Keywords: power mean
zbMATH Keywords: explicit linear differential equations
zbMATH Keywords: integral mean
zbMATH Keywords: geometric mean
reviewed by: János Aczél
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0319
OpenAlex ID: W2078071528
DOI: 10.1006/JMAA.1996.0319
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description: scientific article; zbMATH DE number 930256