Computational Methods and Function Theory 8 (2008), No. 2, 615--624
Copyright Heldermann Verlag 2008
Edward Crane
edward.crane@gmail.com , University of Bristol, Mathematics Department, University Walk, Bristol BS8 1TW, U.K.
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 Ω has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [π2, 4π), thanks to work of Øyma and Rohde. Earlier, Brown Flinn showed that the total length is at most &pi2 in the special case in which L ⊂ Ω. Let r be the anti-Möbius map that fixes $L$ pointwise. In this note we extend the sharp bound &pi2 to the case where each connected component of Ω ∩ r(Ω) is bounded by one arc of ∂Ω and one arc of r(∂Ω). We also strengthen the bounds slightly by replacing Euclidean length with the strictly larger spherical length on |D.
Keywords: Hyperbolic convexity, conformal reflection.
MSC 2000: Primary 30C35; Secondary 30C75, 52A55.
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