On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom
type: Item
instance of: scholarly article
title: On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom
zbMATH Open document ID: 0945.39014
DOI: 10.1007/PL00000129
author: Jean-Luc Marichal
published in: Aequationes Mathematicae
publication date: 2000-10-03T00:00:00Z
review text: \textit{A. N. Kolmogorov} [Atti Accad. Naz. Lincei Rend. (6) 12, 388-391 (1930; JFM 56.0198.02)] and \textit{M. Nagumo} [Japan. J. Math. 7, 71-79 (1930; JFM 56.0198.03)] characterized the quasi-arithmetic means \(M_n(x_1,...,x_n)=f^{-1}(f(x_1)+...f(x_n))\) (\(f: I\to\mathbb{R}\) continuous and strictly monotonic; \(I\) a real interval, \(x_j\in I;\: j=1,\dots,n;\: n=2,3,...\)) by \(M_n (n=2,3,...)\) being reflexive (\(M_n(x,...,x)=x\)), symmetric, continuous, strictly increasing in each variable and unchanged when the first \(k\) variables are replaced by their \(M_k\) mean (\(k=2,...,n-1\)). The author offers the remark that the last property and the symmetry can be replaced by the requirement that \(M_n\) remain unchanged when any \(k\) of its variables is replaced by their \(M_k\) mean.
Mathematics Subject Classification ID: 39B72
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 1436166
zbMATH Keywords: functional equations
zbMATH Keywords: systems
zbMATH Keywords: mean values
zbMATH Keywords: decomposability
zbMATH Keywords: quasiarithmetic means
zbMATH Keywords: JFM 56.0198.02
zbMATH Keywords: JFM 56.0198.03
zbMATH Keywords: symmetry axiom
Wikidata QID: Q114018394
reviewed by: János Aczél
MaRDI profile type: Publication
full work available at URL: pl00000129
OpenAlex ID: W2117818997
description: scientific article; zbMATH DE number 1436166