Abstract
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 π2 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 π2 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.
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References
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Crane, E. A Note on the Hayman-Wu Theorem. Comput. Methods Funct. Theory 8, 615–624 (2008). https://doi.org/10.1007/BF03321708
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DOI: https://doi.org/10.1007/BF03321708