Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Bloch radius, normal families
and quasiregular mappings


Author: Alexandre Eremenko
Journal: Proc. Amer. Math. Soc. 128 (2000), 557-560
MSC (1991): Primary 30C65, 30D45
Published electronically: July 8, 1999
MathSciNet review: 1641689
Full-text PDF Free Access

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. Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
  • 2. Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0357743
  • 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. David Minda, Bloch constants for meromorphic functions, Math. Z. 181 (1982), no. 1, 83–92. MR 671716, 10.1007/BF01214983
  • 9. Ruth Miniowitz, Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc. 84 (1982), no. 1, 35–43. MR 633273, 10.1090/S0002-9939-1982-0633273-X
  • 10. Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 994644
  • 11. Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941
  • 12. Seppo Rickman, The analogue of Picard’s theorem for quasiregular mappings in dimension three, Acta Math. 154 (1985), no. 3-4, 195–242. MR 781587, 10.1007/BF02392472
  • 13. G. Valiron, Recherches sur le théorème de M. Picard, Ann. Sci. École Norm. Sup., 38 (1921), 389-430.
  • 14. Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817. MR 0379852
  • 15. L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc., 35 (1998), 215-230. CMP 98:15

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30C65, 30D45

Retrieve articles in all journals with MSC (1991): 30C65, 30D45


Additional Information

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

DOI: http://dx.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