关于全纯映照模的Schwarz引理一点注记

记DC为单位圆盘,B~k C~k为开欧氏单位球,Ω是C~k(或C)中的域.记H_n(D,Ω)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,∈Hn(D,D),则|f′(z)|≤(n|z|~(n-1))/(1-|z|~(2n))(1-|f|(z|~2),z∈DD同时,对Hn(D,B~k)中映照的模也得到类似的结果.该结论推广了Pavlovic的相应结果.

A Note on a Schwarz Lemma for the Modulus of Holomorphic Mappings

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}.