International Mathematics Competition
for University Students
2021

IMC2021: Day 2, Problem 7

Problem 7. Let \(\displaystyle D \subseteq \mathbb{C}\) be an open set containing the closed unit disk \(\displaystyle \{z\ :\ |z|\le 1\}\). Let \(\displaystyle f:D \to \mathbb{C}\) be a holomorphic function, and let \(\displaystyle p(z)\) be a monic polynomial. Prove that

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

Lars Hörmander

Hint: Apply the maximum principle or the Cauchy formula to a suitable function \(\displaystyle f(z)q(z)\).

  Solution