Abstract
In this paper, the authors establish distortion theorems for various subfamilies H k (\(\Bbb {B}\)) of holomorphic mappings defined in the unit ball in ℂn with critical points, where k is any positive integer. In particular, the distortion theorem for locally biholomorphic mappings is obtained when k tends to +∞. These distortion theorems give lower bounds on | det f′(z)| and Re det f′(z). As an application of these distortion theorems, the authors give lower and upper bounds of Bloch constants for the subfamilies β k (M) of holomorphic mappings. Moreover, these distortion theorems are sharp. When \(\Bbb {B}\) is the unit disk in ℂ, these theorems reduce to the results of Liu and Minda. A new distortion result of Re det f′(z) for locally biholomorphic mappings is also obtained.
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* Project supported by the National Natural Science Foundation of China (No. 10571164), Specialized Research Fund for the Doctoral Program of Higher Education (No. 20050358052) and the Zhejiang Provincial Natural Science Foundation of China (No. Y606197).
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Wang, J., Liu, T. Bloch Constant of Holomorphic Mappings on the Unit Ball of ℂn*. Chin. Ann. Math. Ser. B 28, 677–684 (2007). https://doi.org/10.1007/s11401-006-0433-8
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DOI: https://doi.org/10.1007/s11401-006-0433-8