A lower bound for the Bloch radius of 𝐾-quasiregular mappings

Rajala, Kai

doi:10.1090/S0002-9939-04-07405-2

A lower bound for the Bloch radius of $K$-quasiregular mappings
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Proc. Amer. Math. Soc. 132 (2004), 2593-2601 Request permission

Abstract:

We give a quantitative proof to Eremenko’s theorem (2000), which extends Bloch’s classical theorem to the class of $n$-dimensional $K$-quasiregular mappings.

References

Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
A. Bloch: Les théorèmes de M. Valiron sur les fonctions entières et la théorie de l’uniformisation, Ann. Fac. Sci. Toulouse, 17 (1925).
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 1804531, DOI 10.2307/2661392
A. V. Černavskiĭ, Finite-to-one open mappings of manifolds, Mat. Sb. (N.S.) 65 (107) (1964), 357–369 (Russian). MR 0172256
Andreas Dress, Newman’s theorems on transformation groups, Topology 8 (1969), 203–207. MR 238353, DOI 10.1016/0040-9383(69)90010-X
Alexandre Eremenko, Bloch radius, normal families and quasiregular mappings, Proc. Amer. Math. Soc. 128 (2000), no. 2, 557–560. MR 1641689, DOI 10.1090/S0002-9939-99-05141-2
Juha Heinonen, The branch set of a quasiregular mapping, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 691–700. MR 1957076
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.
O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463, DOI 10.1007/978-3-642-65513-5
O. Martio, U. Srebro, and J. Väisälä, Normal families, multiplicity and the branch set of quasiregular maps, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 1, 231–252. MR 1678028
L. F. McAuley and Eric 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 684659, DOI 10.1090/S0002-9939-1983-0684659-X
David Minda, Bloch constants for meromorphic functions, Math. Z. 181 (1982), no. 1, 83–92. MR 671716, DOI 10.1007/BF01214983
Ruth Miniowitz, Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc. 84 (1982), no. 1, 35–43. MR 633273, DOI 10.1090/S0002-9939-1982-0633273-X
M. H. A. Newman: A theorem on periodic transformations of spaces, Quart. J. Math. Oxford Ser., 2 (1931), 1–9.
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, DOI 10.1090/mmono/073
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, DOI 10.1007/978-3-642-78201-5
G. Valiron: Recherches sur le théorème de M. Picard, Ann. Sci. Ecole Norm. Sup., 38 (1921), 389–430.
Jussi Väisälä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I No. 392 (1966), 10. MR 0200928
Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009, DOI 10.1007/BFb0061216
Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817. MR 379852, DOI 10.2307/2319796