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A Schwarz lemma for meromorphic functions and estimates for the hyperbolic metric
Author(s):
Alexander
Yu.
Solynin
Journal:
Proc. Amer. Math. Soc.
136
(2008),
3133-3143.
MSC (2000):
Primary 30C80
Posted:
May 5, 2008
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Abstract:
We prove a generalization of the Schwarz lemma for meromorphic functions  mapping the unit disk  onto Riemann surfaces  with bounded in mean radial distances from  to the boundary of  . A new variant of the Schwarz lemma is also proved for the Carathèodory class of analytic functions having positive real part in  . Our results lead to several improved estimates for the hyperbolic metric.
References:
-
- 1.
- A. F. Beardon, D. Minda, The hyperbolic metric and geometric function theory. Quasiconformal Mappings and Their Applications, pp. 10-56. Editors: S. Ponnusamy et al., Narosa Publishing House, New Dehi, India, 2007.
- 2.
- R. B. Burckel, D. E. Marshall, D. Minda, P. Poggi-Corradini and T. J. Ransford, Area capacity and diameter versions of Schwarz's lemma. arXiv:0801.3629v1 [math.CV], 23 Jan 2008.
- 3.
- W. K. Hayman, Subharmonic Functions. Vol. 1. London Mathematical Society Monographs, No. 9, Academic Press, Inc., London, 1976. MR 0460672 (57 665)
- 4.
- W. K. Hayman, Subharmonic Functions. Vol. 2. London Mathematical Society Monographs, No. 20, Academic Press, Inc., London, 1989. MR 1049148 (91f:31001)
- 5.
- J. A. Jenkins, Univalent Functions and Conformal Mapping. Second edition, Springer-Verlag, New York, 1965. MR 0096806 (20:3288)
- 6.
- G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Ann. Math. Studies 27, Princeton Univ. Press, 1952. MR 0043486 (13:270d)
- 7.
- A. Yu. Solynin, Some estimates for the capacity of a condenser and the inner radius of a domain (in Russian). Preprint of Kuban State University, Krasnodar, 1983. Deponirovano in VINITI, no. 2015, 18 pp.
- 8.
- A. Yu. Solynin, Solution of a Pólya-Szegö isoperimetric problem. Zap. Nauchn. Semin. LOMI 168 (1988), 140-153; English translation in J. Soviet Math. 53 (1991), 311-320. MR 0982489 (90h:30059)
- 9.
- A. Yu. Solynin, Functional inequalities via polarization. Algebra i Analiz 8 no. 6 (1996), 148-185; English translation in St. Petersburg Math. J. 8, no. 6 (1997), 1015-1038. MR 1458141 (98e:30001a)
- 10.
- A. Yu. Solynin, Moduli and extremal metric problems. Algebra i Analiz 11, no. 1 (1999), 3-86; English translation in St. Petersburg Math. J. 11, no. 1 (2000), 1-65. MR 1691080 (2001b:30058)
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Additional Information:
Alexander
Yu.
Solynin
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409
Email:
alex.solynin@ttu.edu
DOI:
10.1090/S0002-9939-08-09309-X
PII:
S 0002-9939(08)09309-X
Keywords:
Schwarz lemma,
meromorphic function,
hyperbolic metric,
reduced module,
polarization
Received by editor(s):
April 30, 2007
Posted:
May 5, 2008
Additional Notes:
This research was supported in part by NSF grant DMS-0525339
Communicated by:
Ronald A. Fintushel
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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