A refinement of Pawlowski’s result

Let F ( z ) = ∏ k = 1 n ( z − z k ) F(z) = ∏ ₖ₌₁ⁿ(z - z_k) be a monic complex polynomial of degree n n whose zeros satisfy max 1 ≤ k ≤ n | z k | ≤ 1 max 1 ≤ k ≤ n |z_k| ≤ 1 . Pawłowski [Trans. Amer. Math. Soc. 350 (1998), 4461–4472] considered the radius γ n γ _n of the smallest disk, centered at the centroid 1 n ∑ k = 1 n z k 1/n∑ ₖ₌₁^n z_k , containing at least one critical point of F F , establishing the bound γ n ≤ 2 n 1 n − 1 n 2 <mml:mi