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

Abstract

Let Δ be a simply connected domain and f:D→Δ, where D is the unit disk, be a corresponding Riemann map. Let {zn}⊂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(zn)} converges to a point of ∂D by a certain angle θ or by a certain set of angles [θ1,θ2].