The Hayman--Wu constant is $\pi^2$

Abstract

We show that for every conformal map $\phi:\mathbb{D}\to\Omega\subsetneq\mathbb{C}$ from the unit disk onto a simply connected proper domain $\Omega$, the length of $\phi^{-1}(\Omega\cap L)$ is at most $\pi^2$ for every line $L$.