Horocyclically convex univalent functions (Q2491058)

Let \(D\) be the unit disk and \(T=\partial D\). The inner domain of a circle in \(D\) that touches \(D\) is called a horocycle. A domain \(G \subset D\) will be called horocyclically convex (horo-convex) if , for every \(w \in D \cap \partial G\) there exists a horocycle \(H\) such that \[ w \in \partial G \;\;\text{and} \;\;G \cap H = \emptyset . \] A horocyclically convex function f is a conformal map of \(D\) onto a horo-convex domain \(G \subset D\). The aim of this paper is to prove that every horo-convex function is continuous in \(\overline{D}\) and for horo-convex functions \(f\) the paper contains sharp estimates for \(f(\zeta) \) and \(f'(\zeta), \zeta \in T\). Finally, some open problems are given.