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
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