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A Möbius-Invariant Family of Conformal Maps

A Möbius-Invariant Family of Conformal Maps Let f be a conformal map of the unit disk D into Ĉ and let $$Q_{f}(z,\zeta) ={(1-\mid z\mid^2)\mid f'(z)\mid (1-\mid \zeta \mid^2)\mid f'(\zeta) \mid \over \mid f(z) - f(\zeta)\mid^2}\lambda_{\rm D}(z,\zeta)^2$$ , where λD denotes the hyperbolic distance. We introduce the family ML of all conformal maps f for which Q f(z, ζ) remains bounded. It contains all maps f that have a quasi-conformal extension to Ĉ but also some functions for which f(D) has outward-pointing cusps. We show that f has a continuous extension to $\bar {\rm D}$ and study multiple boundary points and the Schwarzian derivative. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

A Möbius-Invariant Family of Conformal Maps

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References (15)

Publisher
Springer Journals
Copyright
Copyright © 2002 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321853
Publisher site
See Article on Publisher Site

Abstract

Let f be a conformal map of the unit disk D into Ĉ and let $$Q_{f}(z,\zeta) ={(1-\mid z\mid^2)\mid f'(z)\mid (1-\mid \zeta \mid^2)\mid f'(\zeta) \mid \over \mid f(z) - f(\zeta)\mid^2}\lambda_{\rm D}(z,\zeta)^2$$ , where λD denotes the hyperbolic distance. We introduce the family ML of all conformal maps f for which Q f(z, ζ) remains bounded. It contains all maps f that have a quasi-conformal extension to Ĉ but also some functions for which f(D) has outward-pointing cusps. We show that f has a continuous extension to $\bar {\rm D}$ and study multiple boundary points and the Schwarzian derivative.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Apr 2, 2013

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