Characterizations of convergence by a given set of angles in simply connected domains

Abstract

Let $\Delta$ be a simply connected domain and $f:\mathbb{D}\to \Delta$, where $\mathbb{D}$ is the unit disk, be a corresponding Riemann map. Let $z_n\subset \Delta$ be a sequence with no accumulation points inside $\Delta$. In the present article, we give necessary and sufficient conditions in terms of hyperbolic geometry which certify that $f^{-1}(z_n)$ converges to a point of $\partial \mathbb{D}$ by a certain angle $\theta$ or by a certain set of angles $[{\theta }_1,{\theta }_2]$.

arXiv comment26 pages, 7 figures