Covering results and perturbed Roper--Suffridge operators

TL;DR

This paper studies advanced Roper--Suffridge extension operators for spirallike functions, introducing perturbed operators that preserve spirallikeness and extending to semigroup generators using a new geometric approach.

Contribution

It introduces perturbed Roper--Suffridge operators that maintain spirallikeness and provides a novel geometric method linking spirallike mappings with semigroups.

Findings

Perturbed operators preserve spirallikeness.

New geometric approach connects spirallike functions with semigroups.

Covering results are crucial for the analysis.

Abstract

This work is devoted to the advanced study of Roper--Suffridge type extension operators. For a given non-normalized spirallike function (with respect to an interior or boundary point) on the open unit disk of the complex plane, we construct perturbed extension operators in a certain class of Banach spaces and prove that these operators preserve the spirallikeness property. In addition, we present an extension operator for semigroup generators. We use a new geometric approach based on the connection between spirallike mappings and one-parameter continuous semigroups. It turns out that the new one-dimensional covering results established below are crucial for our investigation.