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 $\underset{1\le k\le n}{\max }\mathrm{\mid }{z}_k\mathrm{\mid }\le 1$. Pawłowski [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 ${\gamma }_n\le \frac{2n^{\frac{1}{n-1}}}{n^{\frac{2}{n-1}}+1}$. In this paper, inspired by the spirit of Borcea’s variance conjectures and leveraging the classical Schoenberg inequality, we significantly refine Pawłowski’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.