The sum of squared logarithms inequality in arbitrary dimensions

TL;DR

This paper proves the sum of squared logarithms inequality (SSLI) for nonnegative vectors under certain symmetric polynomial conditions, using complex analysis, and explores its applications and broader implications.

Contribution

The paper introduces a novel proof of the SSLI by extending to the complex plane and analyzing derivatives with respect to elementary symmetric polynomials.

Findings

Proved SSLI for nonnegative vectors with symmetric polynomial constraints

Extended the proof to complex variables using complex analysis techniques

Discussed applications and broader connections of the inequality

Abstract

We prove the \emph{sum of squared logarithms inequality} (SSLI) which states that for nonnegative vectors $x, y \in \mathbb{R}^n$ whose elementary symmetric polynomials satisfy $e_k(x)\le e_k(y)$ (for $1\le k < n$) and $e_n(x)=e_n(y)$, the inequality $\sum_i (\log x_i)^2 \le \sum_i (\log y_i)^2$ holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function $f\colon M\subseteq \mathbb{C}^n\to \mathbb{R}$ with $f(z)=\sum_i(\log z_i)^2$ has nonnegative partial derivatives with respect to the elementary symmetric polynomials of $z$. This property leads to our proof. We conclude by providing applications and wider connections of the SSLI.