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References (10)
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On the theorem of Hayman and Wu, 130
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Harmonic Measure, Cambridge new mathematical monographs 2 -
(1983)
Hyperbolic Convexity and Level Sets of Analytic Functions -
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- Publisher
- Springer Journals
- Copyright
- Copyright © Heldermann Verlag 2008
- ISSN
- 1617-9447
- eISSN
- 2195-3724
- DOI
- 10.1007/bf03321708
- Publisher site
- See Article on Publisher Site
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.
Journal
Computational Methods and Function Theory – Springer Journals
Published: May 1, 2008
Keywords: Hyperbolic convexity; conformal reflection; 30C35; 30C75; 52A55