Geometric features of nonlinear resolvents in the unit disk

TL;DR

This paper investigates the geometric properties of nonlinear resolvents of holomorphic generators in the unit disk, establishing distortion, covering, and extension results, and analyzing their implications for semigroup characteristics.

Contribution

It introduces new geometric function theory results for nonlinear resolvents, including distortion, covering, and quasiconformal extension properties, and characterizes their starlikeness.

Findings

Resolved distortion and covering bounds for resolvents

Established order of starlikeness and strong starlikeness

Proved quasiconformal extendability of resolvents

Abstract

We study nonlinear resolvents of holomorphic generators of one-parameter semigroups acting in the open unit disk. The class of nonlinear resolvents can be studied in the framework of geometric function theory because it consists of univalent functions. In this paper we establish distortion and covering results, find order of starlikeness and of strong starlikeness of resolvents. This provides that any resolvent admits quasiconformal extension to the complex plane $\C$. In addition, we obtain some characteristics of semigroups generated by these resolvents.