# Geometric properties of the nonlinear resolvent of holomorphic generators

## TL;DR

This paper investigates the geometric properties of the nonlinear resolvent family associated with holomorphic generators, revealing their convexity, starlikeness, and fixed point characteristics within the unit disk.

## Contribution

It establishes that the resolvent family forms an inverse L"owner chain of hyperbolically convex functions and explores their geometric and fixed point properties.

## Key findings

- Resolvent family forms an inverse L"owner chain of hyperbolically convex functions.
- Each resolvent satisfies the Noshiro-Warschawski condition and is starlike of order at least 1/2.
- Elements of the resolvent family can have quasiconformal extensions and possess repelling fixed points.

## Abstract

Let $f$ be the infinitesimal generator of a one-parameter semigroup $\left\{ F_{t}\right\} _{t\ge0}$ of holomorphic self-mappings of the open unit disk $\Delta$. In this paper we study properties of the family $R$ of resolvents $(I+rf)^{-1}:\Delta\to\Delta~ (r\ge0)$ in the spirit of geometric function theory. We discovered, in particular, that $R$ forms an inverse L\"owner chain of hyperbolically convex functions. Moreover, each element of $R$ satisfies the Noshiro-Warschawski condition and is a starlike function of order at least $\frac12$,. This, in turn, implies that each element of $R$ is also a holomorphic generator. We mention also quasiconformal extension of an element of $R.$ Finally we study the existence of repelling fixed points of this family.