On Kolmogorov-Nagumo averages and nonextensive entropy

By replacing linear averaging in Shannon entropy with Kolmogorov-Nagumo average (KN-average) or quasilinear mean and further imposing the additivity constraint, Rényi proposed the first formal generalization of Shannon entropy. Using this recipe of Rényi, one can prepare only two information measures: Shannon and Rényi entropy. Indeed, using this formalism Rényi characterized these additive entropies in terms of axioms of quasilinear mean. As additivity is a characteristic property of Shannon entropy, pseudo-additivity of the form x ⊕_q y = x + y + (1 - q)xy is a characteristic property of nonextensive (or Tsallis) entropy. One can apply Rényi´s recipe in the nonextensive case by replacing the linear averaging in Tsallis entropy with KN-average and thereby imposing the constraint of pseudo-additivity. In this paper we show that nonextensive entropy is unique under the Rényi´s recipe, and there by give a characterization.