$\mathbb{C}^n$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 $\Omega_{n,p_{2},\cdots,p_{n}}=\{z\in {\mathbb{C}}^{n}:|z_1|^2+|z_2|^{p_2}+\cdots + |z_n|^{p_n}<1\}$, $p_{j}\geq1$, $\beta_{j}\in$ $0, 1]$, $\gamma_{j}\in0, \frac{1}{p_{j}}]$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 the strong starlike mappings of order $\alpha$ as two special classes. The growth and the covering theorems of the subclass of starlike mappings are established. Next, it is proved that the new class is preserved 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 $\Omega_{n,p_{2},\cdots,p_{n}}=\{z\in {\mathbb{C}}^{n}:\,|z_1|^2+|z_2|^{p_2}+\cdots + |z_n|^{p_n}<1\}$, where $p_{j}\geq1$, $\beta_{j}\in0, 1]$, $\gamma_{j}\in0, \frac{1}{p_{j}}]$, such that $\beta_{j}+\gamma_{j}\leq1$, and the branches are chosen such that $\big({\frac{f(z_{1})}{z_{1}}}\big)^{\beta_{j}}\big|_{z_{1}=0}=1$, $(f'(z_{1}))^{\gamma_{j}}\!\!\mid_{z_{1}=0}=1$, $j=2,\cdots,n$. These results enable us to generalize many known results and also lead to some new results.