Computational Methods and Function Theory 10 (2010), No. 1, 207--213
Copyright Heldermann Verlag 2010
A Residual Class of Holomorphic Functions
Charles L. Belna
email@example.com , SoftSolutionz, 410 Mill Pond Drive, Sandusky, OH 44870, U.S.A.
David A. Redett
firstname.lastname@example.org , IPFW, Department of Mathematical Sciences, Fort Wayne, Indiana 46805, U.S.A.
In their 1975 landmark paper, D. D. Bonar and F. W. Carroll have shown that, in the sense of category, there exists a residual class SA of "strongly annular" holomorphic functions in the open unit disk D such that, for each f in SA, there exists an open subset Of,∞ of D such that (1) Of,∞ contains a sequence of concentric circles of increasing radii converging to the unit circle and (2) f(z)→∞ as |z|→ 1 through Of,∞. Because circles have 2-dimensional Lebesgue measure zero, it has been an open question as to whether the set Of,∞ could be chosen to have 2-dimensional measure-theoretic thickness. Here we give a definitive answer to that question. We show that for most functions f in SA, the set Of,∞ can be chosen so that it has upper global metric density 1. Even more, we show that for most functions f in SA and for every complex value ω there exists an open subset Of,ω of D that has upper global metric density $1$ such that $f(z)$ converges to ω as |z|→ 1 through Of,ω.
Keywords: Annular functions, boundary behavior.
MSC 2000: 30D40.
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