IMC 2021 · Problem 7

Let $D \subseteq \mathbb{C}$ be an open set containing the closed unit disk $\{z : |z| \le 1\}$. Let $f \colon D \to \mathbb{C}$ be a holomorphic function, and let $p(z)$ be a monic polynomial. Prove that

$$|f(0)| \le \max_{|z| = 1} |f(z)p(z)|.$$