Invariant and complementary quasi-arithmetic means
type: Item
instance of: scholarly article
title: Invariant and complementary quasi-arithmetic means
zbMATH Open document ID: 0930.26014
published in: Aequationes Mathematicae
publication date: 2000-01-30T00:00:00Z
review text: If \(I\) is a proper (non-singleton) real interval and \(M\) and \(N\) are continuous, both map \(I^{2}\) into \(I\), and both \(M(x,y)\) and \(N(x,y)\) lie between \(\min(x,y)\) and \(\max(x,y),\) one of them always strictly between if \(x\neq y\) (that is, both are means and one of them is a strict mean), then it is easy to see that there exists a unique mean \(K\) such that \(K[M(x,y),N(x,y)]=K(x,y)\) for all \(x,y\) in \(I.\) The problem is more difficult if \(M,N,K\) are all supposed to be quasi-arithmetic, that is, of the form \(f^{-1}([f(x)+f(y)]/2)\) (the continuous bijection \(f\) is the generating function). The author offers the general forms of generating functions of \(M\) and \(N\) if one of them is twice continuously differentiable.
reviewed by: János Aczél
Mathematics Subject Classification ID: 26E60
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 1295532
zbMATH Keywords: functional equations for means
zbMATH Keywords: invariant means
zbMATH Keywords: quasi-arithmetic means
zbMATH Keywords: complementary means
zbMATH Keywords: strict means
author: Janusz Matkowski
MaRDI profile type: Publication
full work available at URL: s000100050072
OpenAlex ID: W2086501405
DOI: 10.1007/S000100050072
description: scientific article; zbMATH DE number 1295532