Why means in two arguments are special
type: Item
instance of: scholarly article
title: Why means in two arguments are special
zbMATH Open document ID: 0860.26014
author: Gunter Lorenzen
published in: Elemente der Mathematik
publication date: 1997-04-21T00:00:00Z
full work available at URL: https://eudml.org/doc/141557
review text: The author notes that both the logarithmic mean \(L(x,y)= (x-y)/(\ln x-\ln y)\) and the geometric mean \(G(x,y)= (xy)^{1/2}\) have interesting applications and both are limits, as \(t\to0\), of one-parameter (and also of two-parameter) mean value families, \[ S(x,y)= t(x^{t+1}- y^{t+1})/((t+1)(x^t-y^t)) \quad\text{and}\quad P(x,y)= ((x^t+y^t)/2)^{1/t}, \] respectively, furthermore, both have analogues in \(n\)-variables, and so have their families, but those of \(G\) and \(P\) seem to be far more interesting than those of \(L\) and \(S\). He attributes this to the fact that the \(n\)-variable analogues of \(P\) and \(G\), the power means \(P_n(x_1,\dots,x_n)= ((x^t_1+\cdots+x^t_n)/n)^{1/t}\) and the geometric mean \(G_n(x_1,\dots,x_n)=(x_1x_2\cdots x_n)^{1/n}\) satisfy, in addition to homogeneity, symmetry, internality (lying between min and max), continuity and strict monotonicity, the Kolmogorov [\textit{A. Kolmogoroff}, Rend. Accad. d.L. Roma (6) 12, 388-391 (1930; JFM 56.0198.02)] -Nagumo [\textit{M. Nagumo}, Jap. J. Math. 7, 71-79 (1930; JFM 56.0198.03)] system of functional equations \[ \begin{multlined} M_n(x_1,\dots,x_r,x_{r+1},\dots,x_n)=\\ M_n(M_r(x_1,\dots,x_r),\dots,M_r(x_1,\dots,x_r),x_{r+1},\dots,x_n) (r=2,\dots, n),\end{multlined}\tag{K-N} \] while no \(n\)-variable generalization of \(L\) or \(S\) can satisfy them since these conditions characterize \(P_n\) and \(G_n\); (K-N), however, is meaningless for \(n=2\) (more exactly, a consequence of internality). \{Note: There exist replacements for (K-N) which, with the other above conditions, characterize \(P_n\) and \(G_n\) and which make also sense for \(n=2\) alone, such as bisymmetry which, for \(n=2\) and under symmetry, postulates that \(M_2(M_2(x,y),M_2(u,v))\) be symmetric in its four variables. If symmetry is dropped then the equation \[ M_2(M_2(x,y),M_2(u,v))= M_2(M_2(x,u),M_2(y,v)) \] is satisfied also by weighted geometric and power means; it seems that the author would consider this too to b…
reviewed by: János Aczél
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 39B62
Mathematics Subject Classification ID: 26B40
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 598843
zbMATH Keywords: Stolarsky means
zbMATH Keywords: Kolmogorov-Nagumo equations
zbMATH Keywords: logarithmic mean
zbMATH Keywords: geometric mean
zbMATH Keywords: power means
zbMATH Keywords: homogeneity
zbMATH Keywords: symmetry
zbMATH Keywords: internality
zbMATH Keywords: continuity
zbMATH Keywords: strict monotonicity
zbMATH Keywords: functional equations
MaRDI profile type: Publication
Recommended article: On certain means of two arguments and their extensions
Recommended article: Q3116233
Recommended article: Inequalities for certain means in two arguments
Recommended article: Q4261105
Recommended article: ON TWO NEW MEANS OF TWO ARGUMENTS III
Recommended article: Some new inequalities for means of two arguments
Recommended article: Q5499622
Recommended article: Q2996338
Recommended article: An Optimal Two Parameter Bounds for the Identric Mean
description: scientific article; zbMATH DE number 598843