ISSN:
1573-8876
Keywords:
singular boundary points of analytic functions
;
Plessner points
;
Fatou points
;
Meyer theorem
;
angular limit
;
logarithmic capacity
;
logarithmic potential
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract This work presents two remarks on the structure of singular boundary sets of functions analytic in the unit diskD: ¦z¦〈1. The first remark concerns the conversion of the Plessner theorem. We prove that three pairwise disjoint subsetsE 1,E 2, andE 3 of the unit circle Γ: ¦z¦=1, $$ \cup _{i = 1}^3 E_i$$ = Γ, are the setsI(ƒ) of all Plessner points,F(ƒ) of all Fatou points, andE(ƒ) of all exceptional boundary points, respectively, for a function ƒ holomorphic inD if and only ifE 1 is aG δ-set andE 3 is a $$G_{\delta \sigma }$$ -set of linear measure zero. In the second part of the paper it is shown that for any $$G_{\delta \sigma }$$ -subsetE of the unit circle Γ with a zero logarithmic capacity there exists a one-sheeted function onD whose angular limits do not exist at the points ofE and do exist at all the other points of Γ.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02316142
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