Matrix-analytic proof of the finite free Stam inequality

Develop a matrix-analytic proof of the finite free Stam inequality for the finite free convolution operator \boxplus_n acting on degree-n real-rooted polynomials, analogous to the derivative-case proof via Gauss–Lucas and differentiator matrices. Concretely, construct a matrix-analytic argument (using the Jacobian of the root map \Omega_{\boxplus_n} that sends root vectors (\alpha, \beta) to the roots of Poly(\alpha) \boxplus_n Poly(\beta)) that establishes the inequality (1/\Phi_n(p)) + (1/\Phi_n(q)) \le 1/\Phi_n(p \boxplus_n q) without relying on hyperbolic-polynomial convexity.

Background

In Section 4 the authors prove finite free Fisher information monotonicity under differentiation using a matrix-analytic approach grounded in explicit formulas for the Jacobian of the derivative root map and its relation to Gauss–Lucas weights and differentiator matrices. This yields sharp bounds via singular-value analysis.

In contrast, the proof of the finite free Stam inequality in Section 6 proceeds through a hyperbolicity-based approach using convexity results of Bauschke, Güler, Lewis, and Sendov, because analogous explicit Jacobian formulas for the finite free convolution root map \Omega_{\boxplus_n} are less tractable. The authors explicitly state that they were unable to apply these formulas to obtain a matrix-analytic proof and suggest it is reasonable to expect such a proof exists, potentially illuminating matrices in finite free position.

This open problem asks for an explicit matrix-analytic proof of the Stam inequality for \boxplus_n, paralleling the derivative-case method, by exploiting the Jacobian of the convolution root map and associated matrix structures.