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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization of a theorem of Heins


Authors: James E. Joseph and Myung H. Kwack
Journal: Proc. Amer. Math. Soc. 128 (2000), 1697-1701
MSC (1991): Primary 32H99; Secondary 30F99, 32H15
Published electronically: September 30, 1999
MathSciNet review: 1641112
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Abstract: Let $\mathcal{H}(\Delta , \Delta )$ be the family of holomorphic selfmaps of the unit disk $\Delta $ in the complex plane $C$. Heins established the continuity of the functional $\psi $ which assigns to $f \in \overline{{\mathcal{H}}(\Delta , \Delta )}-\{id\}$ ($id$ denotes the identity map) either (i) the fixed point of $f$ or (ii) the limit of its iterations or (iii) $f(\Delta )$ if $f(\Delta ) \cap \partial \Delta \not = \emptyset $ ($\partial \Delta $ represents the boundary of $\Delta $). Using an Abate extension of the Denjoy-Wolff lemma to strongly convex domains, we extend this result of Heins to selfmaps of strongly convex domains in $C^{n}$ with $C^{2}$ boundary.


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Additional Information

James E. Joseph
Affiliation: Department of Mathematics, Howard University, Washington, D. C. 20059

Myung H. Kwack
Affiliation: Department of Mathematics, Howard University, Washington, D. C. 20059
Email: mkwack@fac.howard.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05152-7
PII: S 0002-9939(99)05152-7
Keywords: Iterates, fixed points, strongly convex, horosphere
Received by editor(s): February 18, 1998
Received by editor(s) in revised form: July 13, 1998
Published electronically: September 30, 1999
Dedicated: In memory of Professor M. Solveig Espelie
Communicated by: Steven R. Bell
Article copyright: © Copyright 2000 American Mathematical Society