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Abstract: The Hayman-Wu theorem states that the preimage of a line or circle L under aconformal mapping from the unit disc to a simply-connected domain U has totalEuclidean length bounded by an absolute constant. The best possible constant isknown 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 thespecial case in which U contains L. Let r be the anti-M\-obius map that fixes Lpointwise. In this note we extend the sharp bound pi^2 to the case where eachconnected component of the intersection of U with rU is bounded by one arc ofU and its image under r. We also strengthen the bounds slightly by replacingEuclidean length with the strictly larger spherical length restricted to theunit disc.



Author: Edward Crane

Source: https://arxiv.org/



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