| 徐庆华,刘太顺.关于全纯映照模的Schwarz引理一点注记[J].数学年刊A辑,2016,37(2):147~154 |
| 关于全纯映照模的Schwarz引理一点注记 |
| A Note on a Schwarz Lemma for the Modulus of Holomorphic Mappings |
| Received:January 06, 2015 Revised:June 15, 2015 |
| DOI： |
| 中文关键词: Schwarz 引理, $n$阶零点, 全纯映照 |
| 英文关键词:Schwarz lemma, Zero of order $n$, Holomorphic mappings |
| 基金项目:本文受到国家自然科学基金 (No.11561030, No.11261022, No.11471111)和江西省自然科学基金(No.20152ABC20002)的资助. |
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| 中文摘要: |
| 记$\mathbb{D}\subset \mathbb{C}$ 为单位圆盘, $\mathbb{B}^k\subset \mathbb{C}^k$ 为开欧氏单位球, $\Omega$ 是$\mathbb{C}^k$ (或 $\mathbb{C}$) 中的域.
记$H_n(\mathbb{D},\Omega)$为满足一定条件的全纯映照族(或函数族)的全体. 作者证明了若 $f\in H_n(\mathbb{D}, \mathbb{D})$, 则
\begin{align*}
|f'(z)|\leq \frac{n|z|^{n-1}}{1-|z|^{2n}}(1-|f(z)|^2),\quad z\in \mathbb{D}.
\end{align*}
同时, 对$H_n(\mathbb{D}, \mathbb{B}^k)$中映照的模也得到类似的结果. 该结论推广了Pavlovi\'{c}的相应结果. |
| 英文摘要: |
| Let $\mathbb{D}$ be the unit disk in $\mathbb{C}$, $\mathbb{B}^k$ be
the Euclidean unit ball in $\mathbb{C}^k$, $\Omega$ is a domain in
$\mathbb{C}^k$ (or $\mathbb{C}$). Let $H_n(\mathbb{D}, \Omega)$ be
the set of all holomorphic mappings $f$ from $\mathbb{D}$ into
$\Omega$ which satisfies a certain condition. In this paper, it is
proved that if $f\in H_n(\mathbb{D}, \mathbb{D})$, then
\begin{align*}
|f'(z)|\leq \frac{n|z|^{n-1}}{1-|z|^{2n}}(1-|f(z)|^2),\quad z\in
\mathbb{D}.
\end{align*}
Meanwhile, we obtain a similar result for the modulus of mappings
in $H_n(\mathbb{D}, \mathbb{B}^k)$. The result generalizes the
corresponding result obtained earlier by Pavlovi\'{c}. |
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