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The Schwarz-Pick lemma of high order in several variables. (English) Zbl 1208.32001
Let $\Bbb B_n$ denote the unit ball in $\Bbb C^n$, and let $\Omega_{n,m}$ be the class of all holomorphic mappings $f:\Bbb B_n\rightarrow\Bbb B_m$. The authors define the Bergman metric for the unit ball $\Bbb B_n$ as $$ H_n(z;\beta):=\frac{(1-\|z\|^2)\|\beta\|^2+|\langle\beta,z\rangle|^2}{(1-\|z\|^2)^2},\quad z\in\Bbb B^n,\ \beta\in\Bbb C^n, $$ where $\langle\ ,\ \rangle$ denotes the Hermitian scalar product in $\Bbb C^n$ and $\|z\|:=(\langle z,z\rangle)^{1/2}$. For $f\in\Omega_{n,m}$, $k\in\Bbb N$, and $z\in\Bbb B_n$, the Fréchet derivative of $f$ at $z$ of order $k$ is defined by $$ D_k(f,z,\beta):=\sum_{|\alpha|=k}\frac{k!}{\alpha!}\frac{\partial^kf(z)}{\partial z_1^{\alpha_1}\dots\partial z_n^{\alpha_n}}\beta^{\alpha},\quad\beta\in\Bbb C^n. $$ The main result of the paper is the following. Let $f\in\Omega_{n,m}$, $k\in\Bbb N$, $z\in\Bbb B_n$, $\beta\in\Bbb C^n\setminus\{0\}$. Then $$ H_m(f(z);D_k(f,z,\beta))\leqslant(k!)^2\left(1+\frac{|\langle\beta,z\rangle|}{((1-\|z\|^2)\|\beta\|^2+|\langle\beta,z\rangle|^2)^{1/2}}\right)^{2(k-1)}(H_n(z;\beta))^k. $$ It is a generalization of the classical Schwarz-Pick lemma (take $n=m=k=1$) and the result by {\it H. H. Chen} [Sci. China, Ser. A 46, No. 6, 838--846 (2003; Zbl 1097.47509)] (take $k=1$). As a consequence of the main result, the authors obtain a Schwarz-Pick estimate for partial derivatives of a mapping $f\in\Omega_{n,m}$, which, in case $m=1$, is much better than the one obtained by {\it Z. H. Chen} and {\it Y. Liu} [Acta Math. Sin., Engl. Ser. 26, No. 5, 901--908 (2010; Zbl 1243.32002)].

32A10Holomorphic functions (several variables)
32F45Invariant metrics and pseudodistances
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