Bloch constants in several variables
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- by Huaihui Chen and P. M. Gauthier PDF
- Trans. Amer. Math. Soc. 353 (2001), 1371-1386 Request permission
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.References
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
- Huaihui Chen and Paul M. Gauthier, On Bloch’s constant, J. Anal. Math. 69 (1996), 275–291. MR 1428103, DOI 10.1007/BF02787110
- 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
- Carl H. FitzGerald and Sheng Gong, The Bloch theorem in several complex variables, J. Geom. Anal. 4 (1994), no. 1, 35–58. MR 1274137, DOI 10.1007/BF02921592
- Ian Graham and Dror Varolin, Bloch constants in one and several variables, Pacific J. Math. 174 (1996), no. 2, 347–357. MR 1405592
- Kyong T. Hahn, Higher dimensional generalizations of the Bloch constant and their lower bounds, Trans. Amer. Math. Soc. 179 (1973), 263–274. MR 325994, DOI 10.1090/S0002-9947-1973-0325994-2
- Lawrence A. Harris, On the size of balls covered by analytic transformations, Monatsh. Math. 83 (1977), no. 1, 9–23. MR 435454, DOI 10.1007/BF01303008
- Xiang Yang Liu, Bloch functions of several complex variables, Pacific J. Math. 152 (1992), no. 2, 347–363. MR 1141801
- Xiang Yang Liu and David Minda, Distortion theorems for Bloch functions, Trans. Amer. Math. Soc. 333 (1992), no. 1, 325–338. MR 1055809, DOI 10.1090/S0002-9947-1992-1055809-0
- A. Marden and S. Rickman, Holomorphic mappings of bounded distortion, Proc. Amer. Math. Soc. 46 (1974), 226–228. MR 348146, DOI 10.1090/S0002-9939-1974-0348146-5
- E. A. Poletsky, Holomorphic quasiregular mappings, Proc. Amer. Math. Soc. 95 (1985), no. 2, 235–241. MR 801330, DOI 10.1090/S0002-9939-1985-0801330-3
- Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- O. V. Titov, Quasiconformal harmonic mappings of Euclidean space, Dokl. Akad. Nauk SSSR 194 (1970), 521–523 (Russian). MR 0267094
- H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193–233. MR 224869, DOI 10.1007/BF02392083
Additional Information
- Huaihui Chen
- Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097, People’s Republic of China
- Email: hhchen@njnu.edu.cn
- P. M. Gauthier
- Affiliation: Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, Canada H3C 3J7
- Email: gauthier@dms.umontreal.ca
- 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)
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 1371-1386
- MSC (2000): Primary 32H99; Secondary 30C65
- DOI: https://doi.org/10.1090/S0002-9947-00-02734-3
- MathSciNet review: 1806737