Comptes Rendus
no. 7-8
Potential Theory/Complex Analysis
On the behaviour of power series in the absence of Hadamard–Ostrowski gaps
[Sur le comportement des séries entières en lʼabsence de lacunes de Hadamard–Ostrowski]

Présenté par : the Editorial Board

Thomas Kalmes1 ; Jürgen Müller1 ; Markus Nieß2
1 University of Trier, FB IV, Mathematics, Universitätsring, Trier, Germany
2 TU Clausthal, Institute of Mathematics, Erzstr. 1, Clausthal-Zellerfeld, Germany
Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 255-259.

Nous montrons que les sommes partielles (Snf)n∈N dʼune série entière f de rayon de convergence 1 tendent vers ∞ en capacité sur les ensembles compacts (arbitrairement grands) du complémentaire du disque unité fermé si f ne contient pas de lacunes de Hadamard–Ostrowski. Tenant compte dʼun résultat récent de Gardiner, ceci couvre une grande classe de fonctions f holomorphes sur le disque unité.

We show that the partial sums (Snf)n∈N of a power series f with radius of convergence one tend to ∞ in capacity on (arbitrarily large) compact subsets of the complement of the closed unit disk, if f does not have so-called Hadamard–Ostrowski gaps. Regarding a recent result of Gardiner, this covers a large class of functions f holomorphic in the unit disk.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.04.012
Thomas Kalmes 1 ; Jürgen Müller 1 ; Markus Nieß 2

1 University of Trier, FB IV, Mathematics, Universitätsring, Trier, Germany
2 TU Clausthal, Institute of Mathematics, Erzstr. 1, Clausthal-Zellerfeld, Germany 
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Thomas Kalmes; Jürgen Müller; Markus Nieß. On the behaviour of power series in the absence of Hadamard–Ostrowski gaps. Comptes Rendus. Mathématique, Volume 351 (2013) no. 7-8, pp. 255-259. doi : 10.1016/j.crma.2013.04.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.04.012/

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