A refinement of Pawlowski's result

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.