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On a problem of Doob about angular and fine cluster values. (English) Zbl 0593.31005

The main result of this article is the following theorem: Let \(\{p_ n\}\) be a countably dense subset of the unit circle. Then there exists a function f holomorphic on the unit disk and belonging to the Nevanlinna class N such that at each point \(p_ n\) the fine cluster set \(F(p_ n)=\infty\) while the angular cluster set \(A(p_ n)\) is the whole extended complex plane. This theorem sheds some new light in a classical theorem due to J. L. Doob [see Theorem 6.1 in Ann. Inst. Fourier 15, No.1, 113-135 (1965; Zbl 0154.075)] as well as a related problem, also due to Doob, whether F(p) can be a proper subinterval of A(p) on a set of points p of positive measure.
This article also contains a number of further results on angular and fine cluster values. Some of these results are devoted to topological properties of A(p) and F(p). As an example the following might be mentioned: If f is a normal function in the unit disk, then F(p) is both closed and open in A(p) for each point p of the unit circle.
Reviewer: I.Laine

MSC:

31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
30D45 Normal functions of one complex variable, normal families
30D50 Blaschke products, etc. (MSC2000)

Citations:

Zbl 0154.075
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References:

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