On the generalized sum of squared logarithms inequality

TL;DR

This paper characterizes conditions on functions ensuring inequalities involving elementary symmetric polynomials, extending the sum of squared logarithms inequality to higher dimensions and more general functions.

Contribution

It provides necessary and sufficient conditions for a class of inequalities involving symmetric polynomials, generalizing the sum of squared logarithms inequality to larger n and broader functions.

Findings

Established conditions for function f to satisfy the inequality for given symmetric polynomial constraints.

Extended the sum of squared logarithms inequality from n=2,3 to n=4 and beyond.

Identified specific cases where the inequality holds for f(x)=log^2 x and certain subsets S.

Abstract

Assume $n\geq 2$. Consider the elementary symmetric polynomials $e_k(y_1,y_2,\ldots, y_n)$ and denote by $E_0,E_1,\ldots,E_{n-1}$ the elementary symmetric polynomials in reverse order \begin{align*} E_k(y_1,y_2,\ldots,y_n):=e_{n-k}(y_1,y_2,\ldots,y_n)=\sum_{i_1<\ldots<i_{n-k}} y_{i_1}y_{i_2}\ldots y_{i_{n-k}}\, , \quad k\in \{0,1,\ldots,n{-}1 \}\, . \end{align*} Let moreover $S$ be a nonempty subset of $\{0,1,\ldots,n{-}1\}$. We investigate necessary and sufficient conditions on the function $f\colon\,I\to\mathbb{R}$, where $I\subset\mathbb{R}$ is an interval, such that the inequality \begin{align} \label{abstract_inequality} f(a_1)+f(a_2)+\ldots+f(a_n)\leq f(b_1)+f(b_2)+\ldots+f(b_n) \tag{*} \end{align} holds for all $a=(a_1,a_2,\ldots,a_n)\in I^n$ and $b=(b_1,b_2,\ldots,b_n)\in I^n$ satisfying $$E_k(a)< E_k(b) \ \hbox{for } k\in S\quad \hbox{and} \quad E_k(a)=E_k(b) \ \hbox{for } k\in \{0,1,\ldots,n{-}1 \}\setminus S\, .$$ As a corollary, we obtain \eqref{abstract_inequality} if $2\leq n\leq 4$, $f(x)=\log^2x$ and $S=\{1,\dotsc,n-1\}$, which is the sum of squared logarithms inequality previously known for $2\le n\le 3$.