  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.

Complex Variables