An extremal problem for polynomials

Abstract

Let f ( z ) = z n + ⋯ f(z) = {z^n} + \cdots be a polynomial such that the level set E = { z : | f ( z ) | ≤ 1 } E = \{ z:|f(z)| \leq 1\} is connected. Then max { | f ′ ( z ) | : z ∈ E } ≤ 2 ( 1 / n ) − 1 n 2 \max \{ |f’ (z)|:z \in E\} \leq {2^{(1/n) - 1}}{n^2} , and this estimate is the best possible.