On a generalization of sum form functional equation. V
type: Item
instance of: scholarly article
title: On a generalization of sum form functional equation. V
zbMATH Open document ID: 0571.39006
DOI: 10.1007/BF02189418
published in: Aequationes Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137064
review text: [For part IV see Indian J. Math. 22, 257-262 (1980; Zbl 0509.39012).] The following functional equation is considered which is connected with additive measures such as the Shannon entropy, the inaccuracy, the directed divergence, and also with some non-additive ones as the entropy of degree \(\beta\) etc., \[ \sum^{n}_{i=1} \sum^{m}_{j=1} f(x_ i y_ j, u_ i v_ j) = \sum^{n}_{i=1} x^2_ i\;cdot \sum^{m}_{j=1}f(y_ j,v_ j) + \sum^{m}_{j=1} y_ j^{\beta} \cdot \sum^{n}_{i=1}f(x_ i,u_ i), \tag{1} \] for \(U,X\in \Gamma_ n\), \(V,Y\in \Gamma_ m\), \(\alpha\), \(\beta\) non- zero reals. These equations were solved before, when the equations hold for all pairs \(m\), \(n\) or for some particular pairs (2.3), etc. Herein is obtained all the ''measurable'' solutions of (1) holding for some (arbitrary but) fixed pair m,n\(\geq 3:\) Let \(f:J\to\mathbb{R}\) be measurable in each variable. Then f is a solution of (1) for some arbitrary but fixed m,n (\(\geq3\)) if and only if \[ f(x,y) = \begin{cases} a(x^\alpha-x^\beta) + b(x-y),\quad& \alpha\neq\beta \\ ax^\alpha \log x^\alpha + bx \log y + c(x-y),\quad& \alpha =\beta, \quad \alpha\neq1 \text{ for } (x,y)\in J \\ ax\log x + bx\log y + cy\log y + Ax + By + D,\quad& \alpha=\beta=1 \end{cases} \] where \(a\), \(b\), \(c\), \(A\), \(B\) and \(D\) are constants satisfying \(A+B=(mn- m-n)D\).
Mathematics Subject Classification ID: 39B99
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 3910964
zbMATH Keywords: sum form functional equation
zbMATH Keywords: measurable solutions
zbMATH Keywords: additive measures
zbMATH Keywords: Shannon entropy
zbMATH Keywords: inaccuracy
zbMATH Keywords: directed divergence
author: Pl. Kannappan
MaRDI profile type: Publication
cites work: On some functional equations from additive and non-additive measures-I
cites work: Q3921612
cites work: Q4746951
cites work: Q3961247
OpenAlex ID: W3143641103
description: scientific article; zbMATH DE number 3910964