A characterization of orthogonal convergence in simply connected domains

Let $\mathbb{D}$ be the unit disc in $\mathbb{C}$ and let $f:\mathbb D \to \mathbb C$beaRiemannmap,$\Delta=f(\mathbb D)$. We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a compactly divergent sequence $\{z_n\}\subset \mathrm{\Delta }$ 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 $\mathbb{D}$.