Change the coefficients of conditional entropies in extensivity

The Boltzmann--Gibbs entropy is a functional on the space of probability measures. When a state space is countable, one characterization of the Boltzmann--Gibbs entropy is given by the Shannon--Khinchin axioms, which consist of continuity, maximality, expandability and extensivity. Among these four properties, the extensivity is generalized in various ways. The extensivity of a functional is interpreted as the property that, for any random variables $(X,Y)$ taking finitely many values in $\mathbb{N}$, the difference between the value of the functional at the joint law of $(X,Y)$ and that at the law of $X$ coincides with the linear combinations of the values at the conditional laws of $Y$ given $X=n$ with coefficients given by the probabilities of each event $X=n$. A generalization of the extensivity obtained by replacing the coefficients with a power of the probabilities of the events $X=n$ provides a characterization of the Tsallis entropy. In this paper, we first prove the impossibility to replace the coefficients with a non-power function of the probabilities of the events $X=n$. Then we estimate the difference between the value at the joint law of $(X,Y)$ and that at the law of $X$ for a general functional.