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IMC 2021 P7: Maximum in the boundary multiplied by monic polynomial

Source: IMC 2021 P7

August 5, 2021
complex analysisIMC 2021polynomialInequality

Problem Statement

Let D⊆CD \subseteq \mathbb{C} be an open set containing the closed unit disk {z:∣z∣≤1}\{z : |z| \leq 1\}. Let f:D→Cf : D \rightarrow \mathbb{C} be a holomorphic function, and let p(z)p(z) be a monic polynomial. Prove that
∣f(0)∣≤max⁡∣z∣=1∣f(z)p(z)∣ |f(0)| \leq \max_{|z|=1} |f(z)p(z)|
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