A well known result of Beurling asserts that if f is a function which is analytic in the unit disc $\Delta =z\in ℂ:\left|z\right|<1$ and if either f is univalent or f has a finite Dirichlet integral then the set of points $e^{i\theta }$ for which the radial variation $V(f,e^{i\theta })={\int }_0^1\left|f^{\text{'}}\left(r{e}^{i\theta }\right)\right|dr$ 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 $e^{i\theta }$ such that $(1-r)|f^{\text{'}}\left(r{e}^{i\theta }\right)|\ne o\left(1\right)$ as r → 1 is a set of logarithmic capacity zero. In particular, our results give...

The main motivation for this work comes from the century-old Painlevé problem: try to characterize geometrically removable sets for bounded analytic functions in C.