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

Abstract

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