A note on the Hayman-Wu theorem

TL;DR

This paper extends the bounds of the Hayman-Wu theorem on the length of preimages under conformal maps, providing sharper estimates in specific geometric configurations using spherical length.

Contribution

It generalizes the sharp bound of pi^2 to new intersection conditions and improves bounds by considering spherical length instead of Euclidean length.

Findings

Extended the pi^2 bound to new intersection cases.

Replaced Euclidean length with spherical length for stronger bounds.

Provided geometric conditions under which the bounds hold.

Abstract

The Hayman-Wu theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc to a simply-connected domain U has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [pi^2, 4 pi), thanks to work of {\O}yma and Rohde. Earlier, Brown Flinn showed that the total length is at most pi^2 in the special case in which U contains L. Let r be the anti-M\"obius map that fixes L pointwise. In this note we extend the sharp bound pi^2 to the case where each connected component of the intersection of U with r(U) is bounded by one arc of U and its image under r. We also strengthen the bounds slightly by replacing Euclidean length with the strictly larger spherical length restricted to the unit disc.