A refinement of Pawlowski's result

Abstract: Let $F(z)=\prod\limits_{k=1}^n(z-z_k)$ be a monic complex polynomial of degree $n$ with $\max\limits_{1\le k\le n}\left|z_k\right|\le 1$. In 1998, Pawlowski [Trans. Amer. Math. Soc. 350 (1998)] studied the radius $\gamma_n$ of the smallest concentric disk with center at $\tfrac{\sum\limits_{k=1}^nz_k}{n}$ contained at least one critical point of $F(z)$. He showed that $\gamma_n\le \tfrac{2n^\frac{1}{n-1}}{n^\frac{2}{n+1}+1}$. In this paper, we refine Pawlowski's result in the spirit of Borcea variance conjectures and classic Schoenberg inequality, specifically, we show that $\gamma_n\le\sqrt{\tfrac{n-2}{n-1}}$ in a very concise manner. This paper can also be seen as a rare study on the application of Schoenberg inequality.