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Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of $\Bbb C^n$. (English) Zbl 1243.32002
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 $\partial^mf:=\frac{\partial^{|m|}f}{\partial z_1^{m_1}\dots\partial z_n^{m_n}}$, $|m|=\sum_{k=1}^nm_k$.

MSC:
32A10Holomorphic functions (several variables)
32A30Generalizations of function theory to several variables
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30H05Bounded analytic functions
32A05Power series, series of functions (several complex variables)
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References:
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