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Czechoslovak Mathematical Journal, Vol. 72, No. 3, pp. 637-652, 2022

Generic power series on subsets of the unit disk

Balázs Maga, Péter Maga

Received January 21, 2021.   Published online April 20, 2022.   OPEN ACCESS
Abstract:  We examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $\Lambda$ in the sense of Baire category. Notably, we prove that for any open subset $U$ of the unit disk $D$ with a nonreal boundary point on the unit circle, $f(U)$ is a dense set of $\mathbb{C}$. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.
Keywords:  complex power series; boundary behaviour; Baire category
Classification MSC:  30B30, 28A05, 54H05
DOI:  10.21136/CMJ.2022.0021-21
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References:
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Affiliations:   Balázs Maga (corresponding author), Eötvös Loránd University, Department of Analysis, Pázmány Péter sétány 1/C, Budapest, H1117 Hungary, e-mail: mbalazs0701@gmail.com; Péter Maga, Alfréd Rényi Institute of Mathematics, POB 127, Budapest H-1364, Hungary, e-mail: magapeter@gmail.com
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