Distortion Theorem for Bloch Mappings on the Unit Ball ■~n  

Distortion Theorem for Bloch Mappings on the Unit Ball(?)

作　　者：Jian Fei WANG Tai Shun LIU 

机构地区：[1]College of Mathematics and Physics, Information Engineering, Zhejiang Normal University, Jinhua 321004, P. R. China [2]Department of Mathematics, Huzhou Teachers College, Huzhou 313000, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2009年第10期1583-1590,共8页数学学报（英文版）

基　　金：supported by NNSF of China (Grant No.10826083);supported by NNSF of China (Grant No.10571164);NSF of Zhejiang province (Grant No.D7080080);SRFDP of Higher Education (Grant No.20050358052)

摘　　要：In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H（B^n） satisfying ｜｜f｜｜0 = 1 and det f＇（0） = α ∈ （0, 1], where｜｜f｜｜0 = sup{（1 - ｜z｜^2 ）n＋1/2n det（f＇（z））[1/n ： z ∈ B^n}. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f（0）. When a = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara.In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H（B^n） satisfying ｜｜f｜｜0 = 1 and det f＇（0） = α ∈ （0, 1], where｜｜f｜｜0 = sup{（1 - ｜z｜^2 ）n＋1/2n det（f＇（z））[1/n ： z ∈ B^n}. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f（0）. When a = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara.

关 键 词：mapping distortion theorem Bloch mapping univalent ball hyperbolic distance HOLOMORPHIC 

分 类 号：O174[理学—数学] TN722.75[理学—基础数学]