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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hyperbolic lengths of geodesics surrounding two punctures

Authors: Joachim A. Hempel and Simon J. Smith
Journal: Proc. Amer. Math. Soc. 103 (1988), 513-516
MSC: Primary 30C99; Secondary 51M10
MathSciNet review: 943076
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Abstract: For the plane regions $ {\Omega _1} = \left\{ {\left\vert z \right\vert < R,z \ne 0,1} \right\}$ with $ R > 1$, and $ {\Omega _2} = {\mathbf{C}} \setminus \left\{ {0,1,p} \right\}$ with $ \left\vert p \right\vert = R > 1$, we describe, as $ R \to \infty $, the hyperbolic lengths of the geodesies surrounding 0 and 1. Upper and lower bounds for the lengths are also stated, and these results are used to obtain inequalities, which are precise in a certain sense, for the length of the geodesic surrounding 0 and 1 in an arbitrary plane region $ \Omega $ satisfying $ {\Omega _1} \subset \Omega \subset {\Omega _2}$.

References [Enhancements On Off] (What's this?)

  • [1] Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743
  • [2] Zeev Nehari, Conformal mapping, Dover Publications, Inc., New York, 1975. Reprinting of the 1952 edition. MR 0377031

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Article copyright: © Copyright 1988 American Mathematical Society

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