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

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Girela D.

Colloquium Mathematicum

1996

tom 69

nr 1

19-17

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.

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1 Colloquium Mathematicum

1 Girela D.

1 1996

Wyniki wyszukiwania

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