An application of fine potential theory to prove a Phragmen Lindelöf theorem

Lyons, Terry J.

doi:10.5802/aif.964

Lyons, Terry J.

Annales de l'Institut Fourier, Tome 34 (1984) no. 2, pp. 63-66.

Résumé
Abstract

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.

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 $U \supset {| z | > r}$ for some positive $r$ and $f$ has a simple pole.

MR Zbl

DOI : 10.5802/aif.964

@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},
     pages = {63--66},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {34},
     number = {2},
     year = {1984},
     doi = {10.5802/aif.964},
     mrnumber = {86c:30042},
     zbl = {0522.30024},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.964/}
}

TY  - JOUR
AU  - Lyons, Terry J.
TI  - An application of fine potential theory to prove a Phragmen Lindelöf theorem
JO  - Annales de l'Institut Fourier
PY  - 1984
SP  - 63
EP  - 66
VL  - 34
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.964/
DO  - 10.5802/aif.964
LA  - en
ID  - AIF_1984__34_2_63_0
ER  -

%0 Journal Article
%A Lyons, Terry J.
%T An application of fine potential theory to prove a Phragmen Lindelöf theorem
%J Annales de l'Institut Fourier
%D 1984
%P 63-66
%V 34
%N 2
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.964/
%R 10.5802/aif.964
%G en
%F 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, Tome 34 (1984) no. 2, pp. 63-66. doi : 10.5802/aif.964. http://www.numdam.org/articles/10.5802/aif.964/

Bibliographie
Cité par

[1]J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic function, Bull. Soc. Math. France, 85 (1957). | Numdam | MR | Zbl

[2]W. H. J. Fuchs, A Phragmen Lindelöf theorem conjectured by D. J. Newman, T.A.M.S., 257 (1981), 285-293. | MR | Zbl

[3]B. Fuglede, Sur les fonctions finement holomorphes, Ann. Inst. Fourier, Grenoble, 31-4 (1981), 57-88. | Numdam | MR | Zbl

[4]F. W. Gehring, W. K. Hayman and A. Hinkkanen, Analytic functions satisfying a Hölder condition on the boundary, Journal of Approximation Theory, 35 (1982), 243-249. | MR | Zbl

[5]U. Kuran, A new criterion for Dirichlet regularity via quasi-boundedness of the fundamental superharmonic function, J.L.M.S., 19 (1979), 301-311. | MR | Zbl

[6]T. J. Lyons, Finely Holomorphic Functions, and A Theorem in Fine Potential Theory and Applications to Finely Holomorphic Functions, Journ. Functional Analysis, 37 (1980), 1-18 and 19-26. | MR | Zbl

[7]P. A. Meyer, Processus de Markov : la frontière de Martin, Springer Lecture Notes in Mathematics No 77 (1968). | MR | Zbl

Cité par Sources :