A paraconcave entropy function, as defined in [2], is represented by a pair of two real functions of a real variable satisfying certain natural conditions. The subject of this paper is the functional equation, L [ y \\ / (pj)] = y V g(pj), that describes equivalence between two representations of a paraconcave entropy function with / and g satisfying the condition for unbounded entropy. We show that the above equation has a unique solution up to a multiplicative constant in the class of unbounded find complete paraconcave entropy functions. This is a generalization of the result [4] proven already for incomplete paraconcave entropy functions. This is done by invoking a special version of the Cauchy classical functional equation [1]. Additionally, we prove some theorems about ranges of the sums f(j>j) that occur in the investigated equation.

On the functional equation associated with unbounded complete paraconcave entropies

A paraconcave entropy function [2] is represented by a pair of two real functions of a real variable satisfying certain natural conditions. The subject of this paper is the functional equation, ¿ ( y \\ f(pj)) = g(pj), that describes equivalence between two representations of a paraconcave entropy function with concave functions / and g satisfying the condition for a bounded entropy. With the use of E-transforms of the functions / and g we reduce the problem of solvability of the equation to the problem of injectivity of a certain nonlinear operator defined on the set of concave homeomorphisms of the interval [0,1] onto itself. Additionally, we prove some facts about concavity of the E-transform / .

M. Behara

Papers

On paraconcave entropy functions