  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.
