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

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DOI

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. 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
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