Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hyperbolic derivatives and generalized Schwarz-Pick estimates

Authors: Pratibha Ghatage and Dechao Zheng
Journal: Proc. Amer. Math. Soc. 132 (2004), 3309-3318
MSC (2000): Primary 30C80
Published electronically: May 12, 2004
MathSciNet review: 2073307
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we use the beautiful formula of Faa di Bruno for the $n$th derivative of composition of two functions to obtain the generalized Schwarz-Pick estimates. By means of those estimates we show that the hyperbolic derivative of an analytic self-map of the unit disk is Lipschitz with respect to the pseudohyperbolic metric.

References [Enhancements On Off] (What's this?)

  • 1. S. Axler and K. Zhu, Boundary behavior of derivatives of analytic functions, Michigan Math. J. 39 (1992), 129-143. MR 93e:30073
  • 2. L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill, New York, 1973. MR 50:10211
  • 3. A. Beardon, The Schwarz-Pick Lemma for derivatives, Proc. Amer. Math. Soc. 125 (1997), 3255-3256. MR 97m:30062
  • 4. A. Brudnyi, Topology of the maximal ideal space of $H^{\infty}$, J. Funct. Analysis 189 (2002), 21-52. MR 2003c:46066
  • 5. J. Garnett Bounded analytic functions, Academic Press, New York, 1981. MR 83g:30037
  • 6. K. Hoffman, Bounded analytic functions and Gleason parts, Ann. Math. 86 (1967), 74-111. MR 35:5945
  • 7. P. Ghatage, J. Yan and D. Zheng, Composition operators with closed range on the Bloch space, Proc. Amer. Math. Soc. 129 (2001), 2039-2044. MR 2002a:47034
  • 8. B. MacCluer, K. Stroethoff, and R. Zhao, Generalized Schwarz-Pick estimates, Proc. Amer. Math. Soc. 131 (2003), 593-599. MR 2003g:30038
  • 9. K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), 2679-2687. MR 95i:47061
  • 10. A. Montes-Rodríguez, The essential norm of a composition operator on Bloch spaces, Pacific J. Math. 188 (1999), 339-351. MR 2000d:47044
  • 11. S. Ohno, K. Stroethoff and R. Zhao, Weighted composition operators between Bloch-type space, Rocky Mountain J. Math. 33 (2003), 191-215. MR 2004d:47058
  • 12. Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag 299, New York, 1991.MR 95b:30008
  • 13. S. Roman, The Formula of Faa di Bruno, Amer. Math. Monthly 87 (1980), 805-809. MR 82d:26003
  • 14. K. Stroethoff, Lecture notes on The Schwarz-Pick Lemma for derivatives, Preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30C80

Retrieve articles in all journals with MSC (2000): 30C80

Additional Information

Pratibha Ghatage
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115

Dechao Zheng
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Received by editor(s): July 9, 2003
Received by editor(s) in revised form: August 12, 2003
Published electronically: May 12, 2004
Additional Notes: The second author was supported in part by the National Science Foundation.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society