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International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 2, Pages 221-226
http://dx.doi.org/10.1155/S0161171291000236

Inner composition of analytic mappings on the unit disk

Department of Mathematics, University of Southern Colorado, Pueblo, CO 81001-4901, USA

Received 20 June 1989; Revised 29 June 1989

Copyright © 1991 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A basic theorem of iteration theory (Henrici [6]) states that f analytic on the interior of the closed unit disk D and continuous on D with Int(D)f(D) carries any point z ϵ D to the unique fixed point α ϵ D of f. That is to say, fn(z)α as n. In [3] and [5] the author generalized this result in the following way: Let Fn(z):=f1fn(z). Then fnf uniformly on D implies Fn(z)λ, a constant, for all z ϵ D. This kind of compositional structure is a generalization of a limit periodic continued fraction. This paper focuses on the convergence behavior of more general inner compositional structures f1fn(z) where the fj's are analytic on Int(D) and continuous on D with Int(D)fj(D), but essentially random. Applications include analytic functions defined by this process.