SCHWEITZER 2012 Problem 9

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