Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions
Girela, Daniel
Colloquium Mathematicae, Tome 70 (1996), p. 19-17 / Harvested from The Polish Digital Mathematics Library
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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 eiθ for which the radial variation V(f,eiθ)=∫01|f'(reiθ)|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 eiθ such that (1-r)|f'(reiθ)|≠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.

Publié le : 1996-01-01
EUDML-ID : urn:eudml:doc:210320 
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Girela, Daniel. Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions. Colloquium Mathematicae, Tome 70 (1996) pp. 19-17. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv69i1p19bwm/

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