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Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions

Daniel Girela (1996)

Colloquium Mathematicae

A well known result of Beurling asserts that if f is a function which is analytic in the unit disc Δ = z : | z | < 1 and if either f is univalent or f has a finite Dirichlet integral then the set of points e i θ for which the radial variation V ( f , e i θ ) = 0 1 | f ' ( r e i θ ) | d r 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 i θ such that ( 1 - r ) | f ' ( r e i θ ) | o ( 1 ) as r → 1 is a set of logarithmic capacity zero. In particular, our results give...

Some classical function theory theorems and their modern versions

J. L. Doob (1965)

Annales de l'institut Fourier

On étudie les relations entre les valeurs d’adhérence fine en un point-frontière et les valeurs d’adhérence le long de la normale en ce point pour les fonctions sousharmoniques et les fonctions méromorphes dans un demi-plan. Des théorèmes classiques de limite à la frontière et des généralisations sont ainsi obtenues par des méthodes de théorie de potentiel. Une étude de ce genre des valeurs d’adhérence en un point singulier isolé fournit une version en topologie fine du théorème de Casorati-Weierstrass....

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