SCHWEITZER 1959 Problem 9

Let $f(z)= z^n +a_1 z^{n-1}+\dots + a_n$ be a polynomial over the field of the complex numbers and let $E_f$ denote the closed (not necessarily connected) domain of complex numbers $z$ for which $\mid f(z) \mid \leq 1$ . Show that there exists a point $z_0 \in E_f$ such that $\mid f'(z_0) \mid \geq n$ . (F. 5)