Inequalities for rank-two permanents and finite free convolutions

Bang (1976) proved the inequality for matrix permanents ${\mathrm{p}\mathrm{e}\mathrm{r}}^2(A)\ge 2^{-2n}\mathrm{p}\mathrm{e}\mathrm{r}(A\otimes J_2)$, where $J_2$ is the $2\times 2$ all-ones matrix and $A$ is any $n\times n$ matrix with non-negative entries. We show that, if $A$ is any $n\times n$ real-valued matrix with rank at most two (possibly having negative entries), this inequality can be sharpened, replacing the constant $2^{-2n}$ by $1\mathrm{/}\left(\genfrac{}{}{0px}{}{2n}{n}\right)=(n!{)}^2\mathrm{/}(2n)!>2^{-2n}$. We then show that this sharpened inequality also implies new inequalities for finite free convolutions of polynomials: if $p$ and $q$ are monic real-rooted polynomials of degree $n$, then $(p{⊞}_{n}q)(x{)}^2\ge (p^2{⊞}_{2n}{q}^2)(x)$ and $(p{⊠}_{n}q)(x{)}^2\ge (p^2{⊠}_{2n}{q}^2)(x)$ for all $x\in \mathbb{R}$, for ${⊞}_n$ and ${⊠}_n$ the finite free additive and multiplicative convolution operations, respectively, on polynomials of degree $n$.