Three-Parameter Logarithm and Entropy

TL;DR

This paper introduces a new three-parameter logarithmic function that generalizes existing two-parameter versions, explores its properties, and defines a corresponding entropy with desirable stability and concavity features.

Contribution

The paper derives a novel three-parameter logarithm and entropy, extending previous two-parameter models and analyzing their mathematical properties.

Findings

The three-parameter logarithm generalizes the two-parameter version as a limiting case.

The inverse function and key properties of the three-parameter logarithm are established.

The associated three-parameter entropy is shown to be analytic, Lesche-stable, and concave in certain parameter ranges.

Abstract

A three-parameter logarithmic function is derived using the notion of q-analogue and ansatz technique. The derived three-parameter logarithm is shown to be a generalization of the two-parameter logarithmic function of Schwammle and Tsallis as the latter is the limiting function of the former as the added parameter goes to 1. The inverse of the three-parameter logarithm and other important properties are also proved. A three-parameter entropic function is then defined and is shown to be analytic and hence Lesche-stable, concave and convex in some ranges of the parameters.