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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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 
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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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