A note on the Hayman-Wu theorem

Details A note on the Hayman-Wu theorem A note on the Hayman-Wu theorem

2007-07-12

arxiv.org/abs/0707.1772v1

Mathematics

Complex Variables

9 pages, 1 postscript figure. Invited submission to Computational Methods and Function Theory, special issue in honour of Walt

Scientific paper

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.

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