Abstract and Applied Analysis

Volume 2012 (2012), Article ID 647972, 9 pages

doi:10.1155/2012/647972

Research Article

Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball

Jianfei Wang

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.

Abstract.

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.

1. Introduction

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 [1] 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. [2], MacCluer et al. [3], Avkhadiev and Wirths [4], Ghatage and Zheng [5], Dai and Pan [6]) 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. [7] 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 [8] 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 [9].

Let be the infinitesimal form of Carathéodory metric of domain . By the definition of the Carathéodory metric [11], 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 [11]). If , then

In particular, when is biholomorphic mapping, then .

Lemma 2.2 (see [12]). 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

Hence,

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

Thus,

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.

Remark 3.2. If , then the inequality (3.1) reduces to

which coincides with the Theorem 1.1 of Dai and Pan [6] in one complex variable.

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,

That is,

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 [11], 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.

Remark 3.4. If , then and . Thus, the Theorem 3.3 reduces to Theorem A established by Dai et al. [9].

Acknowledgments

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).

References

St. Ruscheweyh, “Two remarks on bounded analytic functions,” Serdica, vol. 11, no. 2, pp. 200–202, 1985.

P. Ghatage, J. Yan, and D. Zheng, “Composition operators with closed range on the Bloch space,” Proceedings of the American Mathematical Society, vol. 129, no. 7, pp. 2039–2044, 2001.

B. D. MacCluer, K. Stroethoff, and R. Zhao, “Generalized Schwarz-Pick estimates,” Proceedings of the American Mathematical Society, vol. 131, no. 2, pp. 593–599, 2003.

F. G. Avkhadiev and K.-J. Wirths, “Schwarz-Pick inequalities for derivatives of arbitrary order,” Constructive Approximation, vol. 19, no. 2, pp. 265–277, 2003.

P. Ghatage and D. Zheng, “Hyperbolic derivatives and generalized Schwarz-Pick estimates,” Proceedings of the American Mathematical Society, vol. 132, no. 11, pp. 3309–3318, 2004.

S. Dai and Y. Pan, “Note on Schwarz-Pick estimates for bounded and positive real part analytic functions,” Proceedings of the American Mathematical Society, vol. 136, no. 2, pp. 635–640, 2008.

J. M. Anderson, M. A. Dritschel, and J. Rovnyak, “Schwarz-Pick inequalities for the Schur-Agler class on the polydisk and unit ball,” Computational Methods and Function Theory, vol. 8, no. 1-2, pp. 339–361, 2008.

Z. H. Chen and Y. Liu, “Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of ,” Acta Mathematica Sinica, vol. 26, no. 5, pp. 901–908, 2010.

S. Dai, H. Chen, and Y. Pan, “The Schwarz-Pick lemma of high order in several variables,” Michigan Mathematical Journal, vol. 59, no. 3, pp. 517–533, 2010.

S. Dai, H. Chen, and Y. Pan, “The high order Schwarz-Pick lemma on complex Hilbert balls,” Science China, vol. 53, no. 10, pp. 2649–2656, 2010.

S. G. Krantz, Function Theory of Several Complex Variables, John Wiley & Sons, New York, NY, USA, 1982.

S. Gong, Convex and Starlike Mappings in Several Complex Variables, vol. 435 of Mathematics and its Applications (China Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.