A note on the definition of deformed exponential and logarithm functions

Abstract

The recent generalizations of the Boltzmann–Gibbs statistics mathematically rely on the deformed logarithmic and exponential functions defined through some deformation parameters. In the present work, we investigate whether a deformed logarithmic/exponential map is a bijection from R+ /R set of positive real numbers/ all real numbers to R/R+, as their undeformed counterparts. We show that their inverse map exists only in some subsets of the aforementioned co domains. Furthermore, we present conditions which a generalized deformed function has to satisfy, so that the most important properties of the ordinary functions are preserved. The fulfillment of these conditions permits us to determine the validity interval of the deformation parameters. We finally apply our analysis to Tsallis q-deformed functions and discuss the interval of concavity of the Rényi entropy