Displaying similar documents to “Generalized John disks”

The Carathéodory topology for multiply connected domains II

Mark Comerford (2014)

Open Mathematics

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We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result...

Short separating geodesics for multiply connected domains

Mark Comerford (2011)

Open Mathematics

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We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which separates these sets and is it a simple closed curve? We show that a shortest geodesic always exists although in general it may not be simple. However, one can also always find a shortest simple curve and we call such a geodesic a meridian of the domain. We...

The Carathéodory topology for multiply connected domains I

Mark Comerford (2013)

Open Mathematics

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We consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge...

The angular distribution of mass by Bergman functions.

Donald E. Marshall, Wayne Smith (1999)

Revista Matemática Iberoamericana

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Let D = {z: |z| < 1} be the unit disk in the complex plane and denote by dA two-dimensional Lebesgue measure on D. For ε > 0 we define Σε = {z: |arg z| < ε}. We prove that for every ε > 0 there exists a δ > 0 such that if f is analytic, univalent and area-integrable on D, and f(0) = 0 then This problem arose in connection with a characterization by Hamilton, Reich and Strebel of extremal dilatation...