Characterizations of generalized entropy functions by functional equations

TL;DR

This paper characterizes a two-parameter extended entropy function and the Tsallis entropy through specific functional equations, providing new insights into their mathematical foundations and properties.

Contribution

It introduces novel functional equations that uniquely characterize the two-parameter extended entropy and Tsallis entropy functions.

Findings

The two-parameter extended entropy is characterized by a specific functional equation.

The Tsallis entropy is characterized by a different functional equation.

An interpretation of the functional equation relates to non-additive properties.

Abstract

We shall show that a two-parameter extended entropy function is characterized by a functional equation. As a corollary of this result, we obtain that the Tsallis entropy function is characterized by a functional equation, which is a different form used in \cite{ST} i.e., in Proposition \ref{prop01} in the present paper. We also give an interpretation of the functional equation giving the Tsallis entropy function, in the relation with two non-additive properties.