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The high order Schwarz-Pick lemma on complex Hilbert balls. (English) Zbl 1211.32001

The aim of the article is to establish a high order Schwarz lemma for holomorphic mappings between the unit balls $𝔹$ and $\stackrel{˜}{𝔹}$ in complex Hilbert spaces $X$ and $Y$, respectively. For $z\in 𝔹$ and $\beta \in X$, we denote by ${H}_{z}\left(\beta ,\beta \right)$ the quantity

${H}_{z}\left(\beta ,\beta \right):=\frac{{\left(1-\parallel z\parallel }^{2}{\right)\parallel \beta \parallel }^{2}+{|〈\beta ,z〉|}^{2}}{{\left(1-\parallel z\parallel }^{2}{\right)}^{2}}·$

For a holomorphic mapping $f:𝔹\to \stackrel{˜}{𝔹}$, we denote by ${D}^{k}f\left(z\right)$ the Fréchet derivative of order $k$ at a point $z\in 𝔹$ and by ${D}^{k}f\left(z\right)·\left({\beta }_{1},\cdots ,{\beta }_{k}\right)$ its evaluation at a $k$-tuple $\left({\beta }_{1},\cdots ,{\beta }_{k}\right)$ of vectors in $X$.

Then the authors’ first main result is as follows:

Theorem 1: Let $f:𝔹\to \stackrel{˜}{𝔹}$ be a holomorphic mapping. Then, for any positive integer $k$, any $z\in 𝔹$ and $\beta \in X\setminus \left\{0\right\}$, one has

${H}_{f\left(z\right)}\left({D}^{k}f\left(z\right)·{\beta }^{k},{D}^{k}f\left(z\right)·{\beta }^{k}\right)\le {\left(k!\right)}^{2}p{\left(z,\beta \right)}^{2\left(k-1\right)}\phantom{\rule{0.166667em}{0ex}}{\left({H}_{z}\left(\beta ,\beta \right)\phantom{\rule{0.166667em}{0ex}}\right)}^{k}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}},$

where ${\beta }^{k}=\left(\beta ,\cdots ,\beta \right)$ and

$p\left(z,\beta \right):=1+\frac{|〈\beta ,z〉|}{\left(\phantom{\rule{0.166667em}{0ex}}\left(1-{\parallel z\parallel }^{2}{\right)\parallel \beta \parallel }^{2}+{|〈\beta ,z〉|}^{2}{\phantom{\rule{0.166667em}{0ex}}\right)}^{1/2}}\phantom{\rule{0.166667em}{0ex}}·$

With the same method of proof as for Theorem 1, the authors obtain

Theorem 2: For holomorphic mappings $f:X\to ℂ$ with a positive real part, one has

$|{D}^{k}f\left(z\right)·{\beta }^{k}|\le 2k!Ref\left(z\right)\phantom{\rule{0.166667em}{0ex}}\right\}p{\left(z,\beta \right)}^{k-1}{H}_{z}{\left(\beta ,\beta \right)}^{k/2}$

for $z\in 𝔹$, an integer $k\ge 1$, and any $\beta \in X\setminus \left\{0\right\}$.

Also, the norm of the $k$-th order Fréchet derivative is estimated in the following two theorems, namely:

Theorem 3: Let $f:𝔹\to \stackrel{˜}{𝔹}$ be a holomorphic mapping. Then for any positive integer $k$ and any $z\in 𝔹$, one has

$\parallel {D}^{k}f\left(z\right)\parallel \le {k}^{k}\sqrt{1-{\parallel f\left(z\right)\parallel }^{2}}\phantom{\rule{0.166667em}{0ex}}\frac{{\left(1+\parallel z\parallel \right)}^{k-1}}{{\left(1-\parallel z\parallel }^{2}{\right)}^{k}}\phantom{\rule{0.166667em}{0ex}}·$

Finally, they obtain for functions as in Theorem 2:

Theorem 4: For holomorphic mappings $f:𝔹\to ℂ$ with a positive real part one has, with integer valued $k\ge 1$, for $z\in 𝔹$ the estimate

$\parallel {D}^{k}f\left(z\right)\parallel \le 2{k}^{k}\left\{Ref\left(z\right)\right\}\frac{{\left(1+\parallel z\parallel \right)}^{k-1}}{{\left(1-\parallel z\parallel }^{2}{\right)}^{k}}·$

##### MSC:
 32A10 Holomorphic functions (several variables) 32F45 Invariant metrics and pseudodistances
##### Keywords:
Hilbert spaces; holomorphic mappings; Schwarz type lemma
##### References:
 [1] Abraham R, Marsden J E, Ratiu T. Manifolds, Tensor Analysis, and Applications, 2nd ed. New York: Springer-Verlag, 1988, 40–116 [2] Chen Z H, Liu Y. Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of $ℂ$n. Acta Math Sinica Engl Ser, 2010, 5: 901–908 · Zbl 1243.32002 · doi:10.1007/s10114-010-7487-y [3] Dai S Y, Chen H H, Pan Y F. The Schwarz-Pick lemma of high order in several variables. Preprint [4] Dai S Y, Pan Y F. Note on Schwarz-Pick estimates for bounded and positive real part analytic functions, Proc Amer Math Soc, 2008, 136: 635–640 · Zbl 1133.30005 · doi:10.1090/S0002-9939-07-09064-8 [5] Maccluer B, Stroethoff K, Zhao R H. Generalized Schwarz-Pick estimates. Proc Amer Math Soc, 2003, 131: 593–599 · Zbl 1012.30015 · doi:10.1090/S0002-9939-02-06588-7 [6] Renaud A. Quelques propriétés des applications analytiques d’une boule de dimension infinie dans une autre. Bull Sci Math, 1973, 97: 129–159 [7] Zhang M Z. Generalized Schwarz-Pick lemma. Acta Math Sinica Chin Ser, 2006, 49: 613–616