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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 \(\mathbb{B}\) and \(\widetilde{\mathbb{B}} \) in complex Hilbert spaces \(X\) and \(Y\), respectively. For \(z \in \mathbb{B}\) and \(\beta \in X\), we denote by \(H_z(\beta,\beta)\) the quantity
\[ H_z(\beta, \beta) := \frac{(1-\|z\|^2)\|\beta\|^2+|\langle \beta ,z \rangle|^2}{(1-\|z\|^2)^2}. \]
For a holomorphic mapping \(f:\mathbb{B} \rightarrow \widetilde{\mathbb{B}}\), we denote by \(D^kf(z)\) the Fréchet derivative of order \(k\) at a point \(z \in \mathbb{B}\) and by \(D^kf(z)\cdot (\beta_1,\dots,\beta_k)\) its evaluation at a \(k\)-tuple \((\beta_1,\dots,\beta_k)\) of vectors in \( X\).
Then the authors’ first main result is as follows:
Theorem 1: Let \(f:\mathbb{B} \rightarrow \widetilde{\mathbb{B}}\) be a holomorphic mapping. Then, for any positive integer \(k\), any \(z \in \mathbb{B}\) and \(\beta \in X \setminus \{0\}\), one has
\[ H_{f(z)} ( D^kf(z)\cdot \beta^k, D^kf(z)\cdot \beta^k ) \leq (k!)^2 p(z,\beta)^{2(k-1)}\,(H_z(\beta,\beta)\,)^k \,\,, \]
where \(\beta^k=(\beta,\dots,\beta)\) and \[ p(z,\beta) :=1 + \frac{|\langle \beta ,z \rangle|}{(\, (1-\|z\|^2)\|\beta\|^2+|\langle \beta ,z \rangle|^2 \,)^{1/2}} \,. \]
With the same method of proof as for Theorem 1, the authors obtain

Theorem 2: For holomorphic mappings \(f:X \rightarrow \mathbb{C}\) with a positive real part, one has
\[ |D^kf(z)\cdot \beta^k| \leq 2k! \operatorname{Re} f(z)\,\} p(z, \beta)^{k-1} H_z(\beta,\beta)^{k/2} \]
for \(z \in \mathbb{B}\), an integer \(k \geq 1 \), and any \(\beta \in X\setminus \{0\}\).
Also, the norm of the \(k\)-th order Fréchet derivative is estimated in the following two theorems, namely:
Theorem 3: Let \(f:\mathbb{B} \rightarrow \widetilde{\mathbb{B}} \) be a holomorphic mapping. Then for any positive integer \(k\) and any \(z \in \mathbb{B}\), one has
\[ \| D^kf(z)\| \leq k^k \sqrt{1-\|f(z)\|^2} \,\frac{(1+\|z\|)^{k-1}}{(1-\|z\|^2)^k}\,. \]
Finally, they obtain for functions as in Theorem 2:
Theorem 4: For holomorphic mappings \(f: \mathbb{B}\rightarrow \mathbb{C}\) with a positive real part one has, with integer valued \(k\geq 1\), for \(z \in \mathbb{B}\) the estimate
\[ \| D^kf(z)\| \leq 2k^k \{\operatorname{Re}f(z) \}\frac{(1+\|z\|)^{k-1}}{(1-\|z\|^2)^k}. \]

MSC:

32A10 Holomorphic functions of several complex variables
32F45 Invariant metrics and pseudodistances in several complex variables
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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 \(\mathbb{C}\)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 · Zbl 1208.32001
[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 · Zbl 0276.32015
[7] Zhang M Z. Generalized Schwarz-Pick lemma. Acta Math Sinica Chin Ser, 2006, 49: 613–616 · Zbl 1115.30022
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