An application of fine potential theory to prove a Phragmen Lindelöf theorem
Annales de l'Institut Fourier, Volume 34 (1984) no. 2, p. 63-66

We give a new proof of a Phragmén Lindelöf theorem due to W.H.J. Fuchs and valid for an arbitrary open subset $U$ of the complex plane: if $f$ is analytic on $U$, bounded near the boundary of $U$, and the growth of $j$ is at most polynomial then either $f$ is bounded or $U\supset \left\{|z|>r\right\}$ for some positive $r$ and $f$ has a simple pole.

On donne une nouvelle démonstration d’un théorème de W.H.J. Fuchs du type Phragmén Lindelöf pour les ouverts $U$ quelconques du plan ouvert : soit $f$ holomorphe dans $U$ et bornée aux environs de la frontière de $U$ croissante ou plus comme un polynôme; alors ou $f$ est bornée ou $f$ a un pôle simple à l’infini.

@article{AIF_1984__34_2_63_0,
author = {Lyons, Terry J.},
title = {An application of fine potential theory to prove a Phragmen Lindel\"of theorem},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Durand},
address = {28 - Luisant},
volume = {34},
number = {2},
year = {1984},
pages = {63-66},
doi = {10.5802/aif.964},
zbl = {0522.30024},
mrnumber = {86c:30042},
language = {en},
url = {http://www.numdam.org/item/AIF_1984__34_2_63_0}
}

Lyons, Terry J. An application of fine potential theory to prove a Phragmen Lindelöf theorem. Annales de l'Institut Fourier, Volume 34 (1984) no. 2, pp. 63-66. doi : 10.5802/aif.964. http://www.numdam.org/item/AIF_1984__34_2_63_0/

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