Uniqueness of Nonextensive entropy under Renyi's Recipe
Abstract
By replacing linear averaging in Shannon entropy with Kolmogorov-Nagumo average (KN-averages) or quasilinear mean and further imposing the additivity constraint, R\'enyi proposed the first formal generalization of Shannon entropy. Using this recipe of R\'enyi, one can prepare only two information measures: Shannon and R\'enyi entropy. Indeed, using this formalism R\'enyi 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)x y is a characteristic property of nonextensive (or Tsallis) entropy. One can apply R\'enyi's recipe in the nonextensive case by replacing the linear averaging in Tsallis entropy with KN-averages and thereby imposing the constraint of pseudo-additivity. In this paper we show that nonextensive entropy is unique under the R\'enyi's recipe, and there by give a characterization.
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