Univalent mappings associated with the Roper-Suffridge extension operator
Let \(B_n\) be the unit ball in \(\mathbb{C}^n\) and \(H(B_n)\) be the set of the holomorphic maps from \(B_n\) into \(\mathbb{C}^n\). If \(f(z)\) is a univalent function of the class \(S\) in the unit disk \(U\in \mathbb{C}^1\) then the Roper-Suffridge operator \({\Phi }_n(f):S\to H(B_n)\) is defined according to the formula \[ {\Phi }_n(f)=(f(z_1),\sqrt{f'(z_1)}z') \] where \(z=(z_1,z')\in \mathbb{C}^n\) with \(z'=(z_2,z_3,\dots,z_n)\). The main result of this paper is the proof of the Roper-Suffridge result that if \(f\) is convex in \(U\) then \({\Phi }_n(f)(z) \) is a convex map of \(B_n\). Some open problems are also discussed.
scientific article; zbMATH DE number 1546766
scientific article