The Roles Played by Order of Starlikeness and the Bloch Condition in the Roper–Suffridge Extension Operator

Abstract

In this article, we obtain that the relation \(\Phi _{G,p}(f)({\mathcal {S}}_1^{*}(\alpha ))\subseteq {\mathcal {S}}_n^{*}\) implies G is a homogeneous polynomial of degree at most p for all \(\alpha \in [0, 1)\) on the bounded convex circular domain \(\Omega _{2,p}\). We shall also show that if f is a spirallike function of type \(\beta \) and is also a Bloch function on the unit disk, then \(\Phi _{G,p}(f)\) is a spirallike mapping of type \(\beta \) on the domain \(\Omega _{2,p}\) for certain G having terms of degree greater than p, where \(p\in {\mathbb {Z}}^+\) and \(p\ge 2\). As corollaries, some well known results can be got.