On linear combinations of weighted quasi-arithmetic means
type: Item
instance of: scholarly article
title: On linear combinations of weighted quasi-arithmetic means
zbMATH Open document ID: 1072.39019
author: Zoltán Daróczy
author: Gabriella Hajdu
published in: Aequationes Mathematicae
publication date: 2005-04-29T00:00:00Z
review text: Let \(CM(I)\) denote the set of all continuous and strictly monotone real functions on the interval \(I\). A mean \(M\) on \(I\) is called a weighted quasi--arithmetic mean if there exists \(\phi \in CM(I)\) such that \[ M(x,y)=\phi^{-1}(\lambda\phi(x)+(1-\lambda)\phi(y))=:A_{\phi}(x,y;\lambda) \qquad (x,y \in I). \] For \(\phi=Id\), \(A_{\phi}(x,y;\lambda)\) is the weighted arithmetic mean and is denoted by \(A(x,y;\lambda)\). The authors consider the following problem. Let \(0<\lambda<1\), \(\mu \neq 0,1\). Find all \(\phi, \psi \in CM(I)\) such that \[ \mu A_{\phi}(x,y;\lambda)+(1-\mu)A_{\psi}(x,y;\lambda)=A(x,y;\lambda) \qquad (x,y \in I). \] After summarizing the known results about this functional equation, they prove an extension theorem for its solutions. The authors claim that this extension theorem can be useful to determine the general solution of the previous equation.
reviewed by: Gian Luigi Forti
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 2162497
zbMATH Keywords: functional equation
zbMATH Keywords: weighted quasi--arithmetic mean
MaRDI profile type: Publication
full work available at URL: s00010-004-2746-6
OpenAlex ID: W2028407387
DOI: 10.1007/S00010-004-2746-6
description: scientific article; zbMATH DE number 2162497