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

Abstract

Let $Δ$ be a simply connected domain and $f:\mathbb{D} \to Δ$, where $\mathbb{D}$ is the unit disk, be a corresponding Riemann map. Let ${z_n}\subset Δ$ be a sequence with no accumulation points inside $Δ$. 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 $θ$ or by a certain set of angles $[θ_1, θ_2]$.