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Zbl 1243.32002
Chen, Zhi Hua; Liu, Yang
Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of $\Bbb C^n$. (English)
[J] Acta Math. Sin., Engl. Ser. 26, No. 5, 901-908 (2010). ISSN 1439-8516; ISSN 1439-7617/e

Let $\Bbb B_n:=\{(z_1,\dots,z_n)\in\Bbb C^n:\sum_{k=1}^n|z_k|^2<1\}$ be the unit ball in $\Bbb C^n$, $\Bbb B_1$ be the unit disc in $\Bbb C$. It is well known, that for any holomorphic function $f:\Bbb B_1\rightarrow\Bbb B_1$ the following Schwarz inequality holds $$ |f'(z)|\leqslant\frac{1-|f(z)|^2}{1-|z|^2},\quad z\in\Bbb B_1. $$ There are many generalizations of this result. One of them is the following one due to {\it S. Y. Dai} and {\it Y. F. Pan} [Proc. Am. Math. Soc. 136, No. 2, 635--640 (2008; Zbl 1133.30005)], which gives the estimate of higher order derivatives: if $f:\Bbb B_1\rightarrow\Bbb B_1$ is holomorphic, then $$ |f^{(m)}(z)|\leqslant\frac{m!(1-|f(z)|^2)}{(1-|z|^2)^m}(1+|z|)^{m-1},\quad m\in\Bbb N,\ z\in\Bbb B_1. $$ The authors generalize this result on several complex variables as follows. Let $f:\Bbb B_n\rightarrow\Bbb B_1$ be holomorphic, $n\in\Bbb N$. Then for any multiindex $m=(m_1,\dots,m_n)\in(\Bbb Z_+^n)_*$ $$ |\partial^mf(z)|\leqslant\binom{n+|m|-1}{n-1}^{n+2}n^{\frac{|m|}{2}}\frac{|m|!(1-|f(z)|^2)}{(1-|z|^2)^{|m|}}(1+|z|)^{|m|-1},\quad z\in\Bbb B_n, $$ where $\partial^mf:=\frac{\partial^{|m|}f}{\partial z_1^{m_1}\dots\partial z_n^{m_n}}$, $|m|=\sum_{k=1}^nm_k$.
[Pawel Zapalowski (Kraków)]

MSC 2000:: *32A10 Holomorphic functions (several variables)
32A30 Generalizations of function theory to several variables
30C80 Maximum principle, etc. (one complex variable)
30H05 Spaces and algebras of analytic functions
32A05 Power series, etc. (several complex variables)

Keywords: Schwarz-Pick estimate; bounded holomorphic function; Taylor expansion

Citations: Zbl 1133.30005

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