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Bloch radius, normal families
and quasiregular mappings


Author: Alexandre Eremenko
Journal: Proc. Amer. Math. Soc. 128 (2000), 557-560
MSC (1991): Primary 30C65, 30D45
DOI: https://doi.org/10.1090/S0002-9939-99-05141-2
Published electronically: July 8, 1999
MathSciNet review: 1641689
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Abstract | References | Similar Articles | Additional Information

Abstract: Bloch's Theorem is extended to $K$-quasiregular maps $\mathbf{R}^n \to\mathbf{S}^n$, where $\mathbf{S}^n$ is the standard $n$-dimensional sphere. An example shows that Bloch's constant actually depends on $K$ for $n\geq 3$.


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

  • 1. L. Ahlfors, Complex Analysis, McGraw-Hill, 1979.MR 80c:30001
  • 2. L. Ahlfors, Conformal Invariants, McGraw-Hill, 1973. MR 50:10211
  • 3. A. Bloch, Ann. Fac. Sci. Toulouse, 17 (1925).
  • 4. S. Bochner, Bloch's theorem for real variables, Bull. Amer. Math. Soc., 52 (1946), 715-719. Collected papers, Part 3, 377-381, AMS, 1992. MR 8:2049
  • 5. M. Bonk and A. Eremenko, Schlicht regions for entire and meromorphic functions, Preprint, 1998.
  • 6. A. V. Chernavskii, Finite-to-one open mappings of manifolds, Mat. Sb., 65 (1964), 357-369, 66 (1964), 471-472. English transl: AMS Transl. (2), 100 (1972).
  • 7. P. Gauthier, Covering properties of holomorphic mappings, to be published in Proc. Int. Conf. Several Compl. Var., Postech, June, 1997, AMS Contemp. Math. Series.
  • 8. D. Minda, Bloch constants for meromorphic functions, Math. Z., 181 (1982), 83-92.MR 84b:30033
  • 9. R. Miniowitz, Normal families of quasimeromorphic mappings, Proc. AMS, 84, 1 (1982), 35-43.MR 83c:30026
  • 10. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Transl. Math. Monogr., Vol 73, AMS, 1989.MR 90d:30067
  • 11. S. Rickman, Quasiregular Mappings, Springer-Verlag, 1993. MR 95g:30026
  • 12. S. Rickman, The analogue of Picard's theorem for quasiregular mappings in dimension three, Acta math., 154 (1985), 195-242. MR 86h:30039
  • 13. G. Valiron, Recherches sur le théorème de M. Picard, Ann. Sci. École Norm. Sup., 38 (1921), 389-430.
  • 14. L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly, 82 (1975), 813-817.MR 52:757
  • 15. L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc., 35 (1998), 215-230. CMP 98:15

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Additional Information

Alexandre Eremenko
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: eremenko@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9939-99-05141-2
Received by editor(s): March 16, 1998
Received by editor(s) in revised form: April 8, 1998
Published electronically: July 8, 1999
Additional Notes: The author was supported by NSF grant DMS-9800084.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1999 American Mathematical Society

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