Mathematical inequalities for some divergences

TL;DR

This paper introduces new parametric divergences combining Jeffreys divergence and Tsallis entropy, establishes inequalities and lower bounds for these divergences, and explores their mathematical properties using Young's inequality.

Contribution

It presents novel parametric extended divergences and derives new inequalities and bounds, advancing the mathematical understanding of divergences in information science.

Findings

New inequalities for extended divergences

Lower bounds for Fermi-Dirac and Bose-Einstein divergences

Inequalities for Tsallis entropy and relative entropy

Abstract

Divergences often play important roles for study in information science so that it is indispensable to investigate their fundamental properties. There is also a mathematical significance of such results. In this paper, we introduce some parametric extended divergences combining Jeffreys divergence and Tsallis entropy defined by generalized logarithmic functions, which lead to new inequalities. In addition, we give lower bounds for one-parameter extended Fermi-Dirac and Bose-Einstein divergences. Finally, we establish some inequalities for the Tsallis entropy, the Tsallis relative entropy and some divergences by the use of the Young's inequality.