## Abstract and Applied Analysis

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

Jianfei Wang

#### Abstract

Let ${B}_{X}$ be the unit ball in a complex Banach space $X$. Assume ${B}_{X}$ is homogeneous. The generalization of the Schwarz-Pick estimates of partial derivatives of arbitrary order is established for holomorphic mappings from the unit ball ${B}^{n}$ to ${B}_{X}$ associated with the Carathéodory metric, which extend the corresponding Chen and Liu, Dai et al. results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 647972, 9 pages.

Dates
First available in Project Euclid: 28 March 2013

https://projecteuclid.org/euclid.aaa/1364475929

Digital Object Identifier
doi:10.1155/2012/647972

Mathematical Reviews number (MathSciNet)
MR2994957

Zentralblatt MATH identifier
1256.32001

#### Citation

Wang, Jianfei. Schwarz-Pick Estimates for Holomorphic Mappings with Values in Homogeneous Ball. Abstr. Appl. Anal. 2012 (2012), Article ID 647972, 9 pages. doi:10.1155/2012/647972. https://projecteuclid.org/euclid.aaa/1364475929

#### 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 ${C}^{n}$,” 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.