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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...

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