On the equation associated with bounded 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, L(sum j f(pj)) = sum j g(pj)i, that describes equivalence between two representations of a paraconcave entropy function with concave functions f and g satisfying the condition for a bounded entropy. With the use of E-transforms of the functions f and g we reduce the problem of solvability of the equation to the problem ofinjectivity of a certain nonlinear operator denned on the set of concave homeomorphisms of the interval [0,1] onto itself. Additionally, we prove some facts about concavity of the E-transform f.