A Note on the Hayman-wu Theorem
نویسنده
چکیده
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 simplyconnected domain Ω has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [π, 4π), thanks to work of Øyma and Rohde. Earlier, Brown Flinn showed that the total length is at most π 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 π 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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تاریخ انتشار 2008