一类多复变全纯映照子族的增长和偏差定理

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.

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

Let $X$ be a complex Banach space with norm $\|\cdot\|$, $B$ the unit ball in $X$. A class of holomorphic mappings $\mathcal{M}_g$ on $B$ is introduced. Let $f(x)$ be a normalized locally biholomorphic mappings on $B$ such that $(Df(x))^{-1}f(x)\in \mathcal{M}_g$ (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 $J_f(z)$ are obtained, where $f(z)$ belongs to the subclasses of starlike mappings defined on the unit polydisc $D^n$ in $\mathbb{C}^n$. These results unify and generalize the corresponding results of many subclasses of starlike mappings.