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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Area, capacity and diameter versions of Schwarz’s Lemma
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by Robert B. Burckel, Donald E. Marshall, David Minda, Pietro Poggi-Corradini and Thomas J. Ransford
Conform. Geom. Dyn. 12 (2008), 133-152
Published electronically: August 27, 2008


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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Bibliographic Information
  • Robert B. Burckel
  • Affiliation: Department of Mathematics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506
  • Email:
  • Donald E. Marshall
  • Affiliation: Department of Mathematics, Box 354350 University of Washington Seattle, Washington 98195-4350
  • MR Author ID: 120295
  • Email:
  • David Minda
  • Affiliation: Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221-0025
  • Email:
  • Pietro Poggi-Corradini
  • Affiliation: Department of Mathematics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506
  • Email:
  • Thomas J. Ransford
  • Affiliation: Département de mathématiques et de statistique, Université Laval, Québec (QC), G1K 7P4, Canada
  • MR Author ID: 204108
  • Email:
  • 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.
  • © Copyright 2008 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 12 (2008), 133-152
  • MSC (2000): Primary 30C80
  • DOI:
  • MathSciNet review: 2434356