Geodesic
Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions
Colloquium Mathematicum, Tome 69 (1996) no. 1, pp. 19-17.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

A well known result of Beurling asserts that if f is a function which is analytic in the unit disc $Δ ={z ∈ ℂ : |z|1} $ and if either f is univalent or f has a finite Dirichlet integral then the set of points $e^{iθ}$ for which the radial variation $V(f,e^{iθ})=∫_{0}^{1}|f'(re^{iθ})|dr$ is infinite is a set of logarithmic capacity zero. In this paper we prove that this result is sharp in a very strong sense. Also, we prove that if f is as above then the set of points $e^{iθ}$ such that $(1 - r)|f'(re^{iθ})| ≠ o(1)$ as r → 1 is a set of logarithmic capacity zero. In particular, our results give an answer to a question raised by T. H. MacGregor in 1983.
DOI : 10.4064/cm-69-1-19-17
Keywords: radial variation, Dirichlet integral, capacity, univalent functions

Daniel Girela 1

1 
@article{10_4064_cm_69_1_19_17,
     author = {Daniel Girela},
     title = {Radial growth and variation of univalent functions and of {Dirichlet} finite holomorphic functions},
     journal = {Colloquium Mathematicum},
     pages = {19--17},
     publisher = {mathdoc},
     volume = {69},
     number = {1},
     year = {1996},
     doi = {10.4064/cm-69-1-19-17},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/cm-69-1-19-17/}
}
TY  - JOUR
AU  - Daniel Girela
TI  - Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions
JO  - Colloquium Mathematicum
PY  - 1996
SP  - 19
EP  - 17
VL  - 69
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4064/cm-69-1-19-17/
DO  - 10.4064/cm-69-1-19-17
LA  - en
ID  - 10_4064_cm_69_1_19_17
ER  - 
%0 Journal Article
%A Daniel Girela
%T Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions
%J Colloquium Mathematicum
%D 1996
%P 19-17
%V 69
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4064/cm-69-1-19-17/
%R 10.4064/cm-69-1-19-17
%G en
%F 10_4064_cm_69_1_19_17
Daniel Girela. Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions. Colloquium Mathematicum, Tome 69 (1996) no. 1, pp. 19-17. doi : 10.4064/cm-69-1-19-17. http://geodesic.mathdoc.fr/articles/10.4064/cm-69-1-19-17/

Cité par Sources :