Equality of two variable Cauchy mean values
type: Item
instance of: scholarly article
title: Equality of two variable Cauchy mean values
zbMATH Open document ID: 1022.39023
author: László Losonczi
published in: Aequationes Mathematicae
publication date: 2003-07-03T00:00:00Z
review text: Given two differentiable real functions \(f,g:I\to \mathbb{R}\) such that \(g'\neq 0\) and \(f'/g'\) is invertible on \(I\), the two variable Cauchy mean \(D_{f,g}\) of two elements \(x\neq y\) of the interval \(I\) is defined as \[ D_{f,g}(x,y):=\left({f'\over g'}\right)^{-1} \left({f(x)-f(y)\over g(x)-g(y)}\right). \] The main result of this paper completely solves the \textit{equality problem} of Cauchy means, i.e., the functional equation \[ D_{f_1,g_1}(x,y)=D_{f_2,g_2}(x,y) \qquad (x,y\in I) \] for seven times continuously differentiable unknown functions \(f_1,g_1:I\to\mathbb{R}\) and \(f_2,g_2:I\to \mathbb{R}\). There are 33 families of solutions. One family, called principal solution, allows two arbitrary functions and four arbitrary constants, while the other 32 families involve only one arbitrary function and several arbitrary constants.
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 1945912
zbMATH Keywords: divided difference
zbMATH Keywords: Cauchy mean value
zbMATH Keywords: equality problem
zbMATH Keywords: functional equation
reviewed by: Zsolt Páles
MaRDI profile type: Publication
full work available at URL: s000100300004
OpenAlex ID: W1550177165
DOI: 10.1007/S000100300004
description: scientific article; zbMATH DE number 1945912