Complete versus incomplete definitions of the deformed logarithmic and exponential functions

TL;DR

This paper investigates the mathematical properties of deformed logarithmic and exponential functions used in generalized statistical mechanics, focusing on their bijectivity, conditions for property preservation, and parameter validity intervals.

Contribution

It provides a detailed analysis of the bijection properties and conditions for deformed functions, clarifying their mathematical validity in generalized statistics.

Findings

Inverse maps exist only on specific subsets.

Conditions for property preservation are established.

Validity intervals for deformation parameters are determined.

Abstract

The recent generalizations of Boltzmann-Gibbs statistics mathematically relies 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 $\mathbb{R}^+/\mathbb{R}$ (set of positive real numbers/all real numbers) to $\mathbb{R}/\mathbb{R}^+$, 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, Kaniadakis, Abe and Borges-Roditi deformed functions.