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A generalization of a theorem of Heins

Authors:	James E Joseph Myung H Kwack

Institution:	Department of Mathematics, Howard University, Washington, D. C. 20059 Myung H. Kwack ; Department of Mathematics, Howard University, Washington, D. C. 20059

Abstract:	Let $\mathcal{H}(\Delta , \Delta )$ be the family of holomorphic selfmaps of the unit disk $\Delta$ in the complex plane . Heins established the continuity of the functional $\psi$ which assigns to $f \in \overline{{\mathcal{H}}(\Delta , \Delta )}-\{id\}$ ( denotes the identity map) either (i) the fixed point of 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.

Keywords:	Iterates fixed points strongly convex horosphere

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