On the sum of squared logarithms inequality and related inequalities

TL;DR

This paper explores the sum of squared logarithms inequality, its connections to majorization theory, and introduces generalized inequalities relevant to information theory, providing new sufficient conditions and extensions.

Contribution

It introduces new sufficient conditions for the sum of squared logarithms inequality and generalizes several inequalities from information theory.

Findings

Established connections between the sum of squared logarithms inequality and majorization.

Derived generalized information and log sum inequalities.

Proposed alternative conditions for vector sets to satisfy the inequalities.

Abstract

We consider the sum of squared logarithms inequality and investigate possible connections with the theory of majorization. We also discuss alternative sufficient conditions on two sets of vectors $a,b\in\mathbb{R}_+^n$ so that $\sum_{i=1}^n(\log a_i)^2\ \leq\ \sum_{i=1}^n(\log b_i)^2\,.\notag $ Generalizations of some inequalities from information theory are obtained, including a generalized information inequality and a generalized log sum inequality, which states for $a,b\in\mathbb{R}_+^n$ and $k_1,...,k_n\in [0,\infty)$: $ \sum_{i=1}^na_i\,\log\prod_{s=1}^m(\frac{a_i}{b_i} + k_s)\ \geq\ \log\prod_{s=1}^m(1+k_s)\,.\notag $