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Zbl 1211.32001
Dai, ShaoYu; Chen, HuaiHui; Pan, YiFei
The high order Schwarz-Pick lemma on complex Hilbert balls.
(English)
[J] Sci. China, Math. 53, No. 10, 2649-2656 (2010). ISSN 1674-7283; ISSN 1869-1862/e

The aim of the article is to establish a high order Schwarz lemma for holomorphic mappings between the unit balls $\bbfB$ and $\widetilde{\bbfB}$ in complex Hilbert spaces $X$ and $Y$, respectively. For $z \in \bbfB$ 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 $f:\bbfB \rightarrow \widetilde{\bbfB}$, we denote by $D^kf(z)$ the Fréchet derivative of order $k$ at a point $z \in \bbfB$ 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 $f:\bbfB \rightarrow \widetilde{\bbfB}$ be a holomorphic mapping. Then, for any positive integer $k$, any $z \in \bbfB$ 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 \smallskip Theorem 2: For holomorphic mappings $f:X \rightarrow \bbfC$ 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 $z \in \bbfB$, 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 $f:\bbfB \rightarrow \widetilde{\bbfB}$ be a holomorphic mapping. Then for any positive integer $k$ and any $z \in \bbfB$, 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 $f: \bbfB\rightarrow \bbfC$ with a positive real part one has, with integer valued $k\geq 1$, for $z \in \bbfB$ the estimate $$\| D^kf(z)\| \leq 2k^k \{\operatorname{Re}f(z) \}\frac{(1+\|z\|)^{k-1}}{(1-\|z\|^2)^k}.$$
[Gregor Herbort (Wuppertal)]
MSC 2000:
*32A10 Holomorphic functions (several variables)
32F45 Invariant metrics and pseudodistances

Keywords: Hilbert spaces; holomorphic mappings; Schwarz type lemma

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