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A lower bound for the Bloch radius of -quasiregular mappings
Author(s):
Kai
Rajala
Journal:
Proc. Amer. Math. Soc.
132
(2004),
2593-2601.
MSC (2000):
Primary 30C65
Posted:
March 25, 2004
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Abstract:
We give a quantitative proof to Eremenko's theorem (2000), which extends Bloch's classical theorem to the class of -dimensional -quasiregular mappings.
References:
-
- 1.
- L. Ahlfors: Conformal Invariants, McGraw-Hill, New York, 1973. MR 50:10211
- 2.
- A. Bloch: Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l'uniformisation, Ann. Fac. Sci. Toulouse, 17 (1925).
- 3.
- M. Bonk and A. Eremenko: Covering properties of meromorphic functions, negative curvature and spherical geometry, Ann. of Math. (2), 152 (2000), no. 2, 551-592. MR 2002a:30050
- 4.
- A. V. Chernavskii: Finite-to-one open mappings of manifolds, Mat. Sb., 65 (1964), 357-369, 66 (1964), 471-472; English transl., Amer. Math. Soc. Transl. (2) 100 (1972), 253-267, 268-270. MR 30:2476; MR 36:3320
- 5.
- A. Dress: Newman's theorem on transformation groups, Topology, 8 (1969), 203-207. MR 38:6629
- 6.
- A. Eremenko: Bloch radius, normal families and quasiregular mappings, Proc. Amer. Math. Soc., 128 (2000), no. 2, 557-560. MR 2000e:30069
- 7.
- J. Heinonen: The branch set of a quasiregular mapping, Proceedings of the ICM Beijing, Higher Education Press, Beijing, 2002, 691-700. MR 2003k:30034
- 8.
- P. Koskela, J. Onninen, and K. Rajala: Mappings of finite distortion: Injectivity radius of a local homeomorphism, Preprint 266, Department of Mathematics and Statistics, University of Jyväskylä, 2002.
- 9.
- O. Lehto and K. I. Virtanen: Quasiconformal mappings in the plane, Springer-Verlag, New York, 1973. MR 49:9202
- 10.
- O. Martio, U. Srebro, and J. Väisälä: Normal families, multiplicity and the branch set of quasiregular mappings, Ann. Acad. Sci. Fenn. Math., 24 (1999), no. 1, 231-252. MR 99m:30043
- 11.
- L. F. McAuley and E. E. Robinson: On Newman's theorem for finite-to-one open mappings on manifolds, Proc. Amer. Math. Soc., 87 (1983), no. 3, 561-566. MR 84d:57007
- 12.
- D. Minda: Bloch constants for meromorphic functions, Math. Z., 181 (1982), 83-92. MR 84b:30033
- 13.
- R. Miniowitz: Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc., 84 (1982), no. 1, 35-43. MR 83c:30026
- 14.
- M. H. A. Newman: A theorem on periodic transformations of spaces, Quart. J. Math. Oxford Ser., 2 (1931), 1-9.
- 15.
- Yu. G. Reshetnyak: Space Mappings with Bounded Distortion, Translations of Mathematical Monographs, vol. 73, Amer. Math. Soc., Providence, RI, 1989. MR 90d:30067
- 16.
- S. Rickman: Quasiregular mappings, Springer-Verlag, New York, 1993. MR 95g:30026
- 17.
- G. Valiron: Recherches sur le théorème de M. Picard, Ann. Sci. Ecole Norm. Sup., 38 (1921), 389-430.
- 18.
- J. Väisälä: Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I, 392 (1966), 1-10. MR 34:814
- 19.
- J. Väisälä: Lectures on
-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin and New York, 1971. MR 56:12260 - 20.
- L. Zalcman: A heuristic principle in complex function theory, Amer. Math. Monthly, 82 (1975), 813-817. MR 52:757
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Additional Information:
Kai
Rajala
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FIN-40014, University of Jyväskylä, Finland
Email:
kirajala@maths.jyu.fi
DOI:
10.1090/S0002-9939-04-07405-2
PII:
S 0002-9939(04)07405-2
Received by editor(s):
April 15, 2003
Received by editor(s) in revised form:
May 23, 2003
Posted:
March 25, 2004
Communicated by:
Juha M. Heinonen
Copyright of article:
Copyright
2004,
American Mathematical Society
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