A characterization of orthogonal convergence in simply connected domains

Abstract: Let $\mathbb D$ be the unit disc in $\mathbb C$ and let $f:\mathbb D \to \mathbb C$ be a Riemann map, $\Delta=f(\mathbb D)$. We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a compactly divergent sequence ${z_n}\subset \Delta$ has the property that ${f^{-1}(z_n)}$ converges orthogonally to a point of $\partial \mathbb D$. We also give some applications of this to the slope problem for continuous semigroups of holomorphic self-maps of $\mathbb D$.