Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of \(\mathbb C^n\)
Let \(\mathbb B_n:=\{(z_1,\dots,z_n)\in\mathbb C^n:\sum_{k=1}^n|z_k|^2<1\}\) be the unit ball in \(\mathbb C^n\), \(\mathbb B_1\) be the unit disc in \(\mathbb C\). It is well known, that for any holomorphic function \(f:\mathbb B_1\rightarrow\mathbb B_1\) the following Schwarz inequality holds \[ |f'(z)|\leqslant\frac{1-|f(z)|^2}{1-|z|^2},\quad z\in\mathbb B_1. \] There are many generalizations of this result. One of them is the following one due to \textit{S. Y. Dai} and \textit{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 \(f:\mathbb B_1\rightarrow\mathbb B_1\) is holomorphic, then \[ |f^{(m)}(z)|\leqslant\frac{m!(1-|f(z)|^2)}{(1-|z|^2)^m}(1+|z|)^{m-1},\quad m\in\mathbb N,\;z\in\mathbb B_1. \] The authors generalize this result on several complex variables as follows. Let \(f:\mathbb B_n\rightarrow\mathbb B_1\) be holomorphic, \(n\in\mathbb N\). Then for any multiindex \(m=(m_1,\dots,m_n)\in(\mathbb Z_+^n)_*\) \[ |\partial^mf(z)|\leqslant\binom{n+|m|-1}{n-1}^{n+2}n^{\frac{|m|}{2}}\frac{|m|!(1-|f(z)|^2)}{(1-|z|^2)^{|m|}}(1+|z|)^{|m|-1},\quad z\in\mathbb B_n, \] where \(\partial^mf:=\frac{\partial^{|m|}f}{\partial z_1^{m_1}\dots\partial z_n^{m_n}}\), \(|m|=\sum_{k=1}^nm_k\).
scientific article; zbMATH DE number 5795520
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