On The Geometry Of Functions Holomorphic In The Unit Circle Of Arbitrarily Slow Growth Which Tend To Infinity On A Sequence Of Curves Approaching The Circumference
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On the Geometry of Functions Holomorphic in the Unit Circle, of Arbitrarily Slow Growth, which Tend to Infinity on a Sequence of Curves Approaching the Circumference
IT IS WELL KNOWN THAT THERE EXIST FUNCTIONS H(), holomorphic in 1, with H() () where (r) is a given positive function which as r 1, and such that min rn H() approaches as n . Here rn 1 is an appropriately chosen sequence. Such functions may be constructed by the use of gap series or via an infinite product. The object of the present note is to construct such a function geometrically by starting with the Riemann surface onto which w = H()maps 1. The essence of the argument is in showing that is hyperbolic and that M(r) (r); these results are obtained via Caratheodory's theory of kernels. (Author).
The Theory of Cluster Sets
Author: E. F. Collingwood
language: en
Publisher: Cambridge University Press
Release Date: 2004-06-03
An introduction to the theory of cluster sets, a branch of topological analysis.