Geometric properties of the nonlinear resolvent of holomorphic generators

Mark Elin, David Shoikhet, Toshiyuki Sugawa

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

Let f be the infinitesimal generator of a one-parameter semigroup of holomorphic self-mappings of the open unit disk Δ. Our main purpose is to study properties of the family R of non-linear resolvents (I+rf)−1:Δ→Δ,r≥0, in the spirit of classical geometric function theory. To make a connection with this theory, we mostly consider the case where f(0)=0 and f(0) is a positive real number. We found, in particular, that R forms an inverse Löwner chain of hyperbolically convex functions. Moreover, each element of R satisfies the Noshiro-Warschawski condition. This, in turn, implies that each element of R is also the infinitesimal generator of a one-parameter semigroup on Δ. We consider also quasiconformal extensions of elements of R. Finally we study the existence of repelling fixed points of this family.

Original languageEnglish
Article number123614
JournalJournal of Mathematical Analysis and Applications
Volume483
Issue number2
DOIs
Publication statusPublished - 2020 Mar 15

Keywords

  • Boundary regular fixed point
  • Hyperbolically convex
  • Inverse Löwner chain
  • Nonlinear resolvent
  • Semigroup generator

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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