A step beyond Tsallis and Rényi entropies

Tsallis and Rényi entropy measures are two possible different generalizations of the Boltzmann-Gibbs entropy (or Shannon's information) but are not generalizations of each others. It is however the Sharma-Mittal measure, which was already defined in 1975 [J. Math. Sci. 10 (1975) 28] and which received attention only recently as an application in statistical mechanics [Physica A 285 (2000) 351, Eur. Phys. J. B 30 (2002) 543] that provides one possible unification. We will show how this generalization that unifies Rényi and Tsallis entropy in a coherent picture naturally comes into being if the q-formalism of generalized logarithm and exponential functions is used, how together with Sharma-Mittal's measure another possible extension emerges which however does not obey a pseudo-additive law and lacks of other properties relevant for a generalized thermostatistics, and how the relation between all these information measures is best understood when described in terms of a particular logarithmic Kolmogorov-Nagumo average.