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An isolated schlicht Function

An isolated schlicht Function Von GEOROE PIRANIAN Let R denote the space of functions (1) /(z) ---- ~ anz" that are holomorphie in the unit disk D, and let the space be metrized by the norm (2) II111 = sup la,,P. Let S denote the subspace of R which consists of those functions (1) that are schlicht in D. tIoRNICR [2] has recently studied the structure of the subspace S; and in a brief note [3], he has exhibited a function / in S with the following property: for some positive number r, none of the functions /(z) + cz (0< Icl < r, c not positive) belongs to S. HORNIC~'S example suggests that the set S may have isolated points; and indeed, by successive modifications of the example, I have arrived at the following proposition. Theorem. There exists a/unction /(z) = ~a,z n (~ [a.I < which belongs to S and lies at a distance one/rom S -- {/}. To prove this theorem, we construct a simply connected domain B, and then we show that every function (1) which maps the unit disk D onto B has the required properties. On the circle [w I ---- 1, we select points wj (j http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

An isolated schlicht Function

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References (5)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02942031
Publisher site
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Abstract

Von GEOROE PIRANIAN Let R denote the space of functions (1) /(z) ---- ~ anz" that are holomorphie in the unit disk D, and let the space be metrized by the norm (2) II111 = sup la,,P. Let S denote the subspace of R which consists of those functions (1) that are schlicht in D. tIoRNICR [2] has recently studied the structure of the subspace S; and in a brief note [3], he has exhibited a function / in S with the following property: for some positive number r, none of the functions /(z) + cz (0< Icl < r, c not positive) belongs to S. HORNIC~'S example suggests that the set S may have isolated points; and indeed, by successive modifications of the example, I have arrived at the following proposition. Theorem. There exists a/unction /(z) = ~a,z n (~ [a.I < which belongs to S and lies at a distance one/rom S -- {/}. To prove this theorem, we construct a simply connected domain B, and then we show that every function (1) which maps the unit disk D onto B has the required properties. On the circle [w I ---- 1, we select points wj (j

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Aug 29, 2008

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