$mathbb{C}^n$中一类星形映射子族的增长定理及推广的Roper-Suffridge算子

在有界星形圆形域上定义了一个新的星形映射子族, 它包含了$alpha$阶星形映射族和$alpha$阶强星形映射族作为两个特殊子类.给出了此类星形映射子族的增长定理和掩盖定理. 另外, 还证明了Reinhardt域$Omega_{n,p_{2},cdots,p_{n}}$上此星形映射子族在Roper-Suffridge算子begin{align*}F(z)=Big(f(z_{1}),Big(frac{f(z_{1})}{z_{1}}Big)^{beta_{2}}(f'(z_{1}))^{gamma_{2}}z_{2},cdots,Big(frac{f(z_{1})}{z_{1}}Big)^{beta_{n}}(f'(z_{1}))^{gamma_{n}}z_{n}Big)'end{align*} 作用下保持不变, 其中$Omega_{n,p_{2},cdots,p_{n}}={zin{mathbb{C}}^{n}:|z_1|^2+|z_2|^{p_2}+cdots + |z_n|^{p_n}<1}$, $p_{j}geq1$, $beta_{j}in$ $[0, 1]$, $gamma_{j}in[0,frac{1}{p_{j}}]$满足$beta_{j}+gamma_{j}leq1$, 所取的单值解析分支使得 $big({frac{f(z_{1})}{z_{1}}}big)^{beta_{j}}big|_{z_{1}=0}=1$,$(f'(z_{1}))^{gamma_{j}}mid_{{z_{1}=0}}=1$, $j=2,cdots,n$. 这些结果不仅包含了许多已有的结果, 而且得到了新的结论.

On the Growth Theorem and the Roper-Suffridge Extension Operator for a Class of Starlike Mappings in ${mathbb{C}}^n$

The author introduces a new subclass of starlike mappings on bounded starlike circular domains, which contains the starlike mappings of order $alpha$ and thestrong starlike mappings of order $alpha$ as two special classes.The growth and the covering theorems of the subclass of starlikemappings are established. Next, it is proved that the new class ispreserved under the following generalized Roper-Suffridge operator:begin{align*}F(z)=Big(f(z_{1}),Big(frac{f(z_{1})}{z_{1}}Big)^{beta_{2}}(f'(z_{1}))^{gamma_{2}}z_{2},cdots,Big(frac{f(z_{1})}{z_{1}}Big)^{beta_{n}}(f'(z_{1}))^{gamma_{n}}z_{n}Big)'end{align*} on Reinhardt domains$Omega_{n,p_{2},cdots,p_{n}}={zin{mathbb{C}}^{n}:,|z_1|^2+|z_2|^{p_2}+cdots + |z_n|^{p_n}<1}$, where$p_{j}geq1$, $beta_{j}in[0, 1]$, $gamma_{j}in[0,frac{1}{p_{j}}]$, such that $beta_{j}+gamma_{j}leq1$, and the branches are chosen such that $big({frac{f(z_{1})}{z_{1}}}big)^{beta_{j}}big|_{z_{1}=0}=1$,$(f'(z_{1}))^{gamma_{j}}!!mid_{z_{1}=0}=1$, $j=2,cdots,n$. Theseresults enable us to generalize many known results and also lead to some new results.