Elementary symmetric polynomials under the fixed point measure

We identify a surprising inequality satisfied by elementary symmetric polynomials under the action of the fixed point measure of a random permutation. Concretely, for any collection of $n$ non-negative real numbers $a_1,\dots ,a_n\in {\mathbb{R}}_{\ge 0}$, we prove that $$\frac{1}{n!}\sum _{\pi \in S_n}\left[\prod _{\{i:i=\pi (i)\}}{a}_i\right]\ge \frac{1}{\left(\genfrac{}{}{0px}{}{n}{2}\right)}\sum _{S\in \left(\genfrac{}{}{0px}{}{[n]}{2}\right)}\left[{\left(\prod _{\{i\in S\}}{a}_i\right)}^{1\mathrm{/}2}\right],$$ and this bound is sharp. To prove this elementary inequality, we construct a collection of differential operators to set up a monotone flow that then allows us to establish the inequality.