Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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

  • [A1973] Ahlfors, L. V. Conformal invariants: topics in geometric function theory. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, 1973. MR 0357743 (50:10211)
  • [Cara1907] C. Carathéodory, Sur quelques applications du théorème de Landau-Picard, C. R. Acad. Sci. Paris, 144 (1907), 1203-1206.
  • [Carl1921] T. Carleman, Zur Theorie der Minimalflächen, Math. Z. 9 (1921), 154-160. MR 1544458
  • [D1983] P. Duren, Univalent functions, Springer-Verlag, 1983. MR 708494 (85j:30034)
  • [EG1992] L. Evans and R. Gariepy, Measure theory and fine properties of functions, CRC Press, 1992. MR 1158660 (93f:28001)
  • [F1942] W. Feller, Some geometric inequalities, Duke Math. J., 9 (1942), 885-892. MR 0007622 (4:168g)
  • [Gam2001] T. Gamelin, Complex Analysis, Springer, 2001. MR 1830078 (2002h:30001)
  • [GeH1999] F.W. Gehring and K. Hag, Hyperbolic geometry and disks, J. Comput. Appl. Math., 105 (1999) 275-284. MR 1690594 (2000e:30085)
  • [LaT1907] E. Landau and O. Toeplitz, Über die größte Schwankung einer analytischen Funktion in einem Kreise, Arch. der Math. und Physik (3) 11 (1907), 302-307.
  • [Lax1995] P. Lax, A short path to the shortest path Amer. Math. Monthly, 102 (1995), n. 2, 158-159. MR 1315595
  • [Li1919] L. Lichtenstein, Neuere Entwicklung der Potentialtheorie. Konforme Abbildung, Encyklopädie der Mathematischen Wissenschaften Bd.II, 3rd Part. 1st Half, Heft 3, 181-377. B.G.Teubner (1919), Leipzig.
  • [Mac1964] T. MacGregor, Length and area estimates for analytic functions, Michigan Math. J. 11 (1964), 317-320. MR 0171003 (30:1236)
  • [MinW1982] D. Minda and D. Wright, Univalence criteria and hyperbolic metric, Rocky Mountain J. Math. 12 (1982) no. 3, 471-479. MR 672231 (84e:30023)
  • [Pol1928] G. Pólya, Beitrag zur Verallgemeinerung des Verzerrungssatzes auf merhfach zusammenhängende Gebiete II, S.-B. Preuss. Akad. (1928), 280-282.
  • [PolS1951] G. Pólya and G. Szegő, Isoperimetric inequalities in mathematical physics, Princeton Univ. Press, 1951. MR 0043486 (13:270d)
  • [PolS1972] G. Pólya and G. Szegő, Problems and Theorems in Analysis, Vol. 1, Springer-Verlag, 1972. MR 0344042 (49:8782)
  • [Pou1907] K. A. Poukka, Über die größte Schwankung einer analytischen Funktion auf einer Kreisperipherie, Arch. der Math. und Physik (3) 12 (1907), 251-254.
  • [Ra1995] T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Student Texts 28, Cambridge Univ. Press, Cambridge, 1995. MR 1334766 (96e:31001)
  • [Re1991] R. Remmert, Theory of Complex Functions, Graduate Texts in Math., vol. 122, Springer-Verlag, 1991. MR 1084167 (91m:30001)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30C80

Retrieve articles in all journals with MSC (2000): 30C80

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

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

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

American Mathematical Society