A sum form functional equation on a closed domain and its role in information theory

TL;DR

This paper finds general solutions to a specific sum form functional equation involving probability distributions, with implications for information theory, extending previous work by Losonczi and Maksa.

Contribution

It provides a comprehensive solution to a complex functional equation relevant to information theory, generalizing earlier special cases.

Findings

Derived general solutions for the functional equation.

Extended previous results by Losonczi and Maksa.

Highlighted the equation's role in information theory.

Abstract

This paper is devoted to finding the general solutions of the functional equation $\sumin \sumjm h(p_iq_j)=\sumin h(p_i)+\sumjm k_j(q_j)+\lambda\sumin h(p_i)\sumjm k_j(q_j)$ valid for all complete probability distributions $(p_1,\ldots,p_n)$, $(q_1,\ldots,q_m)$, $0\le p_i\le 1$, $0\le q_j\le 1$, $i=1,\ldots,n$; $j=1,\ldots,m$, $\sumin p_i=1$, $\sumjm q_j=1$; $n\ge 3$, $m\ge 3$ fixed integers; $\lambda\in\RR$, $\lambda\neq 0$ and the mappings $h:I\to\RR$, $k_j:I\to\RR$, $j=1,\ldots,m$; $I=[0,1]$, $\RR$ denoting the set of all real numbers. A special case of the above functional equation was treated earlier by L. Losonczi and Gy. Maksa.