Extreme points, support points and \(g\)-Loewner chains associated with Roper-Suffridge and Pfaltzgraff-Suffridge extension operators
In this paper the author considers an extension operator of the Pfaltzgraff-Suffridge type, denoted by \(\Psi_{n,\alpha}\), where \(\alpha\geq 0\), and obtains a subordination preserving result with respect to this operator. This result is contained in Theorem 2.1. Certain particular cases of Theorem 2.1 are also obtained. The main results of this paper are contained in Theorems 3.1 and 3.2, regarding extreme points and support points associated with the compact family \(\Psi_{n,\alpha}(\overline{S_g^0(B^n)})\), where \(S_g^0(B^n)\) is the family of normalized univalent mappings with \(g\)-parametric representation on the Euclidean unit ball \(B^n\), and \(g\) is a univalent function on the unit disc \(U\) such that \(g(0)=1\), \(g(\overline{\zeta})=\overline{g(\zeta)}\) for \(\zeta\in U\), \(\Re g(\zeta)>0\), \(\zeta\in U\), and \(g\) satisfies certain natural assumptions. In the last section, the author obtains examples of \(g\)-star-like mappings, \(g\)-spiral-like mappings of type \(\alpha\), and \(g\)-almost star-like mappings of order \(\alpha\) on \(B^n\), and uses these notions to obtain some preservation results under certain extension operators.
Pfaltzgraff-Suffridge extension operator
scientific article; zbMATH DE number 6506175
scientific article