Lengths of radii under conformal maps of the unit disc

If $E_{f}(R)$ is the set of endpoints of radii which have length greater than or equal to $R>0$ under a conformal map $f$ of the unit disc, then $\operatorname{cap} E_{f}(R)=O(R^{-1/2})$ as $R\to \infty$ for the logarithmic capacity of $E_{f}(R)$. The exponent $-1/2$ is sharp.