一类多复变全纯映照子族的增长和偏差定理

Growth and Distortion Theorems for a Subclass of Holomorphic Mappings in Several Complex Variables

在一般复Banach空间X中的单位球B上引入一类全纯映照族M_g.考虑B上满足条件(Df(x))^(-1)f(x)∈M_g的正规化局部双全纯映照f(x)(其中x=0是f(x)-x的k+1阶零点)并得到其增长定理.作为应用,也得到了C^n中单位多圆柱D^n上映照f关于Jacobi矩阵Jf(z)的偏差定理,该结果统一和推广了星形映照许多子族的相应结论.

Let X be a complex Banach space with norm ｜｜·｜｜, B the unit ball in X. A class of holomorphic mappings Mg on B is introduced. Let f（x） be a normalized locally biholomorphic mappings on B such that （Df（x））^-1f（x） ∈Mg （where x = 0 is the zero of order k＋1 of f（x）-x）. The authors investigate the growth theorem for f（x）. As applications, the distortion theorems for the Jacobian matrix Jf（z） are obtained, where f（z） belongs to the subclasses of starlike mappings defined on the unit polydisc D^n in C^n. These results unify and generalize the corresponding results of many subclasses of starlike mappings.