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Bloch constants in several variables

Authors: Huaihui Chen and P. M. Gauthier
Journal: Trans. Amer. Math. Soc. 353 (2001), 1371-1386
MSC (2000): Primary 32H99; Secondary 30C65
Published electronically: December 18, 2000
MathSciNet review: 1806737
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Abstract: We give lower estimates for Bloch's constant for quasiregular holomorphic mappings. A holomorphic mapping of the unit ball $B^n$ into $\mathbf{C}^n$ is $K$-quasiregular if it maps infinitesimal spheres to infinitesimal ellipsoids whose major axes are less than or equal to $K$ times their minor axes. We show that if $f$ is a $K$-quasiregular holomorphic mapping with the normalization $\det f'(0) =1,$ then the image $f(B^n)$contains a schlicht ball of radius at least $1/12K^{1-1/n}.$ This result is best possible in terms of powers of $K.$ Also, we extend to several variables an analogous result of Landau for bounded holomorphic functions in the unit disk.

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Additional Information

Huaihui Chen
Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097, People’s Republic of China

P. M. Gauthier
Affiliation: Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, Canada H3C 3J7

Keywords: Bloch constant
Received by editor(s): August 10, 1998
Published electronically: December 18, 2000
Additional Notes: Research supported in part by NSFC(China), NSERC(Canada) and FCAR(Québec)
Article copyright: © Copyright 2000 American Mathematical Society

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