Geometric properties of the nonlinear resolvent of holomorphic generators

2019/01/08 by Mark Elin, Elin, Mark, David Shoikhet +3 · 1 citation
Engineering · Mathematics · #30C62 #47H10 #Analytic and geometric function theory #Complex Variables (math.CV) #Elasticity and Wave Propagation #FOS: Mathematics #Holomorphic and Operator Theory #Primary 30C80 #Secondary 30C45

paper · pdf · doi:10.48550/arxiv.1901.02142

openalex publication_date 2019/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f be the infinitesimal generator of a one-parameter semigroup \ F_t\ _t≥0 of holomorphic self-mappings of the open unit disk Δ. In this paper we study properties of the family R of resolvents (I+rf)^-1:Δ→Δ~ (r≥0) in the spirit of geometric function theory. We discovered, in particular, that R forms an inverse Löwner 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.

Citations

• On starlike and convex functions of order α

Cited by

• Hyperbolic curvature of holomorphic level curves

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