A refinement of Pawlowski's result

Abstract

Let \(F(z) = \prod_{k=1}^{n}(z - z_k)\) be a monic complex polynomial of degree \(n\) whose zeros satisfy \(\max\limits_{1 \le k \le n} |z_k| \le 1\). Paw{\l}owski [Trans. Amer. Math. Soc. 350(11) (1998)] considered the radius \(\gamma_n\) of the smallest disk, centered at the centroid \(\frac{1}{n}\sum_{k=1}^n z_k\), containing at least one critical point of \(F\), establishing the bound $\gamma_n \le \frac{2\,n^{\frac{1}{n-1}}}{n^{\frac{2}{n-1}} + 1}$. In this paper, inspired by the spirit of Borcea's variance conjectures and leveraging the classical Schoenberg inequality, we significantly refine Paw{\l}owski's estimate by proving succinctly and elegantly that $\gamma_n \le \sqrt{\frac{n - 2}{n - 1}}$. This result also represents a rare and noteworthy application of Schoenberg's inequality to the geometry of polynomial critical points.