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Miklós Schweitzer 2012 P9

Source: Miklós Schweitzer 2012 P9

August 20, 2018
college contestsMiklos Schweitzer

Problem Statement

Let DD be the complex unit disk D={z∈C:∣z∣<1}D=\{z \in \mathbb{C}: |z|<1\}, and 0<a<10<a<1 a real number. Suppose that f:D→C∖{0}f:D \to \mathbb{C}\setminus \{0\} is a holomorphic function such that f(a)=1f(a)=1 and f(−a)=−1f(-a)=-1. Prove that sup⁡z∈D∣f(z)∣⩾exp⁡(1−a24aπ). \sup_{z \in D} |f(z)| \geqslant \exp\left(\frac{1-a^2}{4a}\pi\right) .
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