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The Schwarz-Pick lemma of high order in several variables. (English) Zbl 1208.32001

Let ${𝔹}_{n}$ denote the unit ball in ${ℂ}^{n}$, and let ${{\Omega }}_{n,m}$ be the class of all holomorphic mappings $f:{𝔹}_{n}\to {𝔹}_{m}$. The authors define the Bergman metric for the unit ball ${𝔹}_{n}$ as

${H}_{n}\left(z;\beta \right):=\frac{{\left(1-\parallel z\parallel }^{2}{\right)\parallel \beta \parallel }^{2}+{|〈\beta ,z〉|}^{2}}{{\left(1-\parallel z\parallel }^{2}{\right)}^{2}},\phantom{\rule{1.em}{0ex}}z\in {𝔹}^{n},\phantom{\rule{4pt}{0ex}}\beta \in {ℂ}^{n},$

where $〈\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}〉$ denotes the Hermitian scalar product in ${ℂ}^{n}$ and $\parallel z\parallel :={\left(〈z,z〉\right)}^{1/2}$.

For $f\in {{\Omega }}_{n,m}$, $k\in ℕ$, and $z\in {𝔹}_{n}$, the Fréchet derivative of $f$ at $z$ of order $k$ is defined by

${D}_{k}\left(f,z,\beta \right):=\sum _{|\alpha |=k}\frac{k!}{\alpha !}\frac{{\partial }^{k}f\left(z\right)}{\partial {z}_{1}^{{\alpha }_{1}}\cdots \partial {z}_{n}^{{\alpha }_{n}}}{\beta }^{\alpha },\phantom{\rule{1.em}{0ex}}\beta \in {ℂ}^{n}·$

The main result of the paper is the following.

Let $f\in {{\Omega }}_{n,m}$, $k\in ℕ$, $z\in {𝔹}_{n}$, $\beta \in {ℂ}^{n}\setminus \left\{0\right\}$. Then

${H}_{m}\left(f\left(z\right);{D}_{k}\left(f,z,\beta \right)\right)⩽{\left(k!\right)}^{2}{\left(1+\frac{|〈\beta ,z〉|}{\left(\left(1-{\parallel z\parallel }^{2}{\right)\parallel \beta \parallel }^{2}+{|〈\beta ,z〉|}^{2}{\right)}^{1/2}}\right)}^{2\left(k-1\right)}{\left({H}_{n}\left(z;\beta \right)\right)}^{k}·$

It is a generalization of the classical Schwarz-Pick lemma (take $n=m=k=1$) and the result by 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 Z. H. Chen and Y. Liu [Acta Math. Sin., Engl. Ser. 26, No. 5, 901–908 (2010; Zbl 05795520)].

##### MSC:
 32A10 Holomorphic functions (several variables) 32F45 Invariant metrics and pseudodistances
##### Keywords:
Schwarz-Pick lemma; Bergman metric