A note on the Hayman-Wu theorem

The Hayman-Wu theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc D to a simply-connected domain Omega 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 Oyma and Rohde. Earlier, Brown Flinn showed that the total length is at most pi^2 in the special case in which Omega contains L. Let r be the anti-Mobius 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 Omega with r(Omega) is bounded by one arc of the boundary of Omega and its image under r. We also strengthen the bounds slightly by replacing Euclidean length with the strictly larger spherical length on D.