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Two hyperbolic Schwarz lemmas

Published online by Cambridge University Press:  17 April 2009

L. Bernal-González  and
M. C. Calderón-Moreno
L. Bernal-González
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Apdo. 1160, Avenida Reina Mercedes, 41080 Sevilla, Spain, e-mail: lbernal@us.es
M. C. Calderón-Moreno
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Apdo. 1160, Avenida Reina Mercedes, 41080 Sevilla, Spain, e-mail: mccm@us.es
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Abstract

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In this paper, a sharp version of the Schwarz–Pick Lemma for hyperbolic derivatives is provided for holomorphic selfmappings on the unit disk with fixed multiplicity for the zero at the origin. This extends a recent result due to Beardon. A property of preserving hyperbolic distances also studied by Beardon is here completely characterised.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 1 , August 2002 , pp. 17 - 24
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Beardon, A.F., ‘The Schwarz–Pick Lemma for derivatives’, Proc. Amer. Math. Soc. 125 (1997), 3255–3256.CrossRefGoogle Scholar
[2]Dieudonné, J., ‘Recherches sur quelques problèmes relatifs aux polynômes et aux fonctions bornées d'une variable complexe’, Ann. Sci. Ecole Norm. Sup. 48 (1931), 247–358.CrossRefGoogle Scholar
[3]Donoghe, W.F. Jr., Distributions and Fourier transforms (Academic Press, New York, 1966).Google Scholar