MathDB

Let

D

be the complex unit disk

D=\{z \in \mathbb{C}: |z|<1\}

, and

0<a<1

a real number. Suppose that

f:D \to \mathbb{C}\setminus \{0\}

is a holomorphic function such that

f(a)=1

and

f(-a)=-1

. Prove that

\sup_{z \in D} |f(z)| \geqslant \exp\left(\frac{1-a^2}{4a}\pi\right) .

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