Lesche stability of the Shannon strong hyperbolic entropy and some hyperbolic extensions

Abstract

In recent decades, several definitions of new entropy measures have been proposed, which expands the range of applications for this important tool. The present work focuses on the extension of the classical Shannon entropy to the hyperbolic number plane $\mathbb{D}$ with the notion of valued hyperbolic probability. It is shown that the Shannon strong hyperbolic entropy over a discrete hyperbolic probability distribution $(\rho_{1}, \ldots,\rho_{N})$ can be established by the action of the hyperbolic derivative on the generating function $\sum_{s=1}^{N}{\rho}_{s}^{-\xi}$ with respect to the hyperbolic variable $\xi$ and then we tend $\xi$ to $-1_{\mathbb{D}}$. Furthermore, we prove that this hyperbolic extension possesses the Lesche stability property, also known as experimental robustness. Finally, we present some results on the hyperbolic extension of the R\'enyi entropy and hyperbolic extropy.