On a functional equation related to a generalization of Flett's mean value theorem
type: Item
instance of: scholarly article
title: On a functional equation related to a generalization of Flett's mean value theorem
zbMATH Open document ID: 0949.39009
DOI: 10.1155/S0161171200000508
author: Thomas Riedel
author: Maciej Sablik
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2000-11-29T00:00:00Z
full work available at URL: https://eudml.org/doc/48572
review text: The authors offer a proof, without regularity assumptions, that the general solutions \(f: \mathbb{R} \to\mathbb{R}\) of the functional equation (\(q\in ]0,1[\) constant) \[ f((1-q)x+qy)-f(x)=q(y-x)h((1-q)x+qy)-(h(y)-h(x))q^2(y-x)/2 \] (\((x,y)\in\mathbb{R}^2, x\neq y\)) are quadratic polynomials for arbitrary \(q\in]0,1[\) and, only if \(q=3/4,\) also cubic polynomials. The second unknown function, \(h\), turns out to be the derivative of \(f\), thus establishing a connection to \textit{T. M. Flett}'s mean value theorem [Math. Gaz. 42, 38-39 (1958)]. [Remark: The trivial case \(a=0\) (\(x=0\) in our notation), neglected in the lines after eqs. (2.18) and (2.22), can easily be taken care of by substituting \(a=0\) into eqs. (2.17) and (2.21), respectively].
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 1441870
zbMATH Keywords: functional equation with two unknown functions
zbMATH Keywords: mean value theorem
reviewed by: János Aczél
MaRDI profile type: Publication
OpenAlex ID: W2010248523
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description: scientific article; zbMATH DE number 1441870