Refined Young inequality and its application to divergences

TL;DR

This paper presents refined bounds on the difference between weighted means, leading to improved Young inequalities and their reverses, with applications to various divergence measures in information theory.

Contribution

It introduces refined Young inequalities and explores their properties, applying these results to analyze divergences like Tsallis, Rényi, and Jensen-Shannon-Tsallis.

Findings

Established bounds on mean differences

Derived refined Young inequalities and reverses

Applied inequalities to divergence measures

Abstract

We give bounds on the difference between the weighted arithmetic mean and the weighted geometric mean. These imply refined Young inequalities and the reverses of the Young inequality. We also study some properties on the difference between the weighted arithmetic mean and the weighted geometric mean. Applying the newly obtained inequalities, we show some results on the Tsallis divergence, the R\'{e}nyi divergence, the Jeffreys-Tsallis divergence and the Jensen-Shannon-Tsallis divergence.