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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 647972, 9 pages
Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball
Department of Mathematics, Zhejiang Normal University, Zhejiang, Jinhua 321004, China
Received 11 July 2012; Accepted 21 October 2012
Academic Editor: Abdelghani Bellouquid
Copyright © 2012 Jianfei Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be the unit ball in a complex Banach space . Assume is homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ball to associated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.
By the classical Pick’s invariant form of Schwarz’s lemma, a holomorphic function which is bounded by one in the unit disk satisfies the following inequlity at each point of D. Ruscheweyh in  firstly obtained best-possible estimates of higher order derivatives of bounded holomorphic functions on the unit disk in 1985. Recently, a lot of attention (see Ghatage et al. , MacCluer et al. , Avkhadiev and Wirths , Ghatage and Zheng , Dai and Pan ) has been paid to the Schwarz-Pick estimates of high-order derivative estimates in one complex variable. The best result is given as follows:
It is natural to consider an extension of the above Schwarz-Pick estimates to higher dimensions. Anderson et al.  gave Schwarz-Pick estimates of derivatives of arbitrary order of functions in the Schur-Agler class on the unit polydisk and the unit ball of , respectively. Recently, Chen and Liu in  obtained estimates of high-order derivatives for all the bounded holomorphic functions on the unit ball of . Later, Dai et al. in [9, 10] generalized the high order Schwarz-Pick estimates for holomorphic mappings between unit balls in complex Hilbert space. Their main result is expressed as follows.
Theorem A. Suppose is holomorphic mapping from to . Then for any multiindex and , where and is the Bergman metric on .
In this paper, we will extend Theorem A to holomorphic mappings from the unit ball to associated with the Carathéodory metric. In particular, when , our result coincides with Theorem A. Furthermore, our result shows that the high-order Schwarz-Pick estimates on the unit ball do depend on the geometric property of the image domain .
Throughout this paper, the symbol is used to denote a complex Banach space with norm , and is the unit ball in . Let be the space of complex variables with the Euclidean inner product , where the symbol ’ stands for the transpose of vector or matrix. The unit ball of is always written by . It is well known that if is a holomorphic mapping from into , then the following well-known expansion holds for all in some neighborhood of , where means the th Fréchet derivative of at the point , and Furthermore, is a bounded symmetric -linear mapping from into . For a domain , a mapping is called to be biholomorphic if is a domain; the inverse exists and is holomorphic on . Let denote the set of biholomorphic mappings of onto itself. is said to be homogeneous, if for each pair of points , there is an such that .
In multiindex notation, is an -tuple of nonnegative integers, , , .
Let be the Bergman kernel function. Then the Bergman metric can be defined as where . It is well known that in .
Let be the infinitesimal form of Carathéodory metric of domain . By the definition of the Carathéodory metric , we have for any , where denotes the family of holomorphic mappings which map into .
2. Some Lemmas
In order to prove the main results, we need the following lemmas. Let be the unit ball in a complex Banach space , and is homogeneous.
Lemma 2.1 (see ). If , then In particular, when is biholomorphic mapping, then .
Lemma 2.2 (see ). Consider the following:
Lemma 2.3. Let . Then can be written with the following -variable power series given by Then the following holds for any integer .
Proof. For the fixed , we define
Then . It is clear that
From the power series expansion of the holomorphic function , we get
In terms of the homogeneity of , we can take and , then . This implies that
By making use of the orthogonality, we obtain
This implies the following inequality
holds for any . Thus,
holds for any . It means that .
By Lemmas 2.1 and 2.2, we obtain which is the desired result.
3. Main Results
Theorem 3.1. Let be a holomorphic mapping. Then the following inequality holds for and .
Proof. Let be a holomorphic function on defined by
Then can be written as a power series as follows:
In order to obtain Theorem 3.1, we need to prove the following equality:
Let such that , the Cauchy integral formula shows that
Let . Then
Substituting (3.7) into (3.6), we get
which prove the equality (3.4).
From Lemma 2.3, we have for any integer , This implies that which completes the desired result.
Theorem 3.3. Let be a holomorphic mapping. Then the following inequality holds for , and .
Proof. For any fixed , and . Define the following disk:
Notice that . Hence,
Set . For the fixed and , we define
Then is holomorphic mapping from the unit disk to the homogeneous domain .
According to Theorem 3.1 to the functions and , we have which holds for . Since , and In terms of the chain rule, we have Hence, Note the definition of Carathéodory metric and in , we can get This gives the proof of the case and . For general vector , we may substitute for . By the homogeneous of from the above inequality, we can obtain the same result, which completes the proof of the Theorem 3.3.
The author cordially thanks the referees’ thorough reviewing with useful suggestions and comments made to the paper. The author would also like to express this appreciation to Dr. Liu Yang for giving him some useful discussions. This work was supported by the National Natural Science Foundation of China (nos. 11001246, 11101139), NSF of Zhejiang province (nos. Y6110260, Y6110053, and LQ12A01004), and Zhejiang Innovation Project (no. T200905).
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