The sum of squared logarithms inequality in arbitrary dimensions

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.