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Conformal Geometry and Dynamics

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Area, capacity and diameter versions of Schwarz’s Lemma

Authors: Robert B. Burckel, Donald E. Marshall, David Minda, Pietro Poggi-Corradini and Thomas J. Ransford
Journal: Conform. Geom. Dyn. 12 (2008), 133-152
MSC (2000): Primary 30C80
Published electronically: August 27, 2008
MathSciNet review: 2434356
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Abstract: The now canonical proof of Schwarz’s Lemma appeared in a 1907 paper of Carathéodory, who attributed it to Erhard Schmidt. Since then, Schwarz’s Lemma has acquired considerable fame, with multiple extensions and generalizations. Much less known is that, in the same year 1907, Landau and Toeplitz obtained a similar result where the diameter of the image set takes over the role of the maximum modulus of the function. We give a new proof of this result and extend it to include bounds on the growth of the maximum modulus. We also develop a more general approach in which the size of the image is estimated in several geometric ways via notions of radius, diameter, perimeter, area, capacity, etc.

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Additional Information

Robert B. Burckel
Affiliation: Department of Mathematics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506

Donald E. Marshall
Affiliation: Department of Mathematics, Box 354350 University of Washington Seattle, Washington 98195-4350
MR Author ID: 120295

David Minda
Affiliation: Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221-0025

Pietro Poggi-Corradini
Affiliation: Department of Mathematics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506

Thomas J. Ransford
Affiliation: Département de mathématiques et de statistique, Université Laval, Québec (QC), G1K 7P4, Canada
MR Author ID: 204108

Received by editor(s): July 17, 2007
Published electronically: August 27, 2008
Additional Notes: The second author was supported by NSF grant DMS 0602509.
The fifth author was supported by grants from NSERC, FQRTN, and the Canada research chairs program.
Article copyright: © Copyright 2008 American Mathematical Society