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Nonlinear resolvents: distortion and order of starlikeness

Mark Elin

TL;DR

The paper addresses the geometric structure of nonlinear resolvents of holomorphically accretive mappings on the open unit ball of a complex Banach space. It develops a distortion theorem to bound resolvent norms, proves that nonlinear resolvents are holomorphically accretive with a quantified squeezing ratio, and establishes that resolvents are starlike of at least order $\tfrac{1}{2}$ with sharper, parameter-dependent refinements. The results rely on representing generators via $f(x)=p(x)x$, employing the Riesz–Herglotz framework, and analyzing the resolvent equation $G_\lambda(x)+\lambda p(G_\lambda(x))G_\lambda(x)=x$ to obtain precise estimates. Collectively, these findings extend one- and finite-dimensional resolvent theory to the Banach-space setting, yielding concrete bounds and conditions for accretivity and starlikeness that illuminate the dynamics of the associated holomorphic semigroups and their geometric behavior.

Abstract

This paper presents a new approach to studying nonlinear resolvents of holomorphically accretive mappings on the open unit ball of a complex Banach space. We establish a distortion theorem and apply it to address problems in geometric function theory concerning the class of resolvents. Specifically, we prove the accretivity of resolvents and provide estimates for the squeezing ratio. Further, we show that nonlinear resolvents are starlike mappings of certain order and determine lower bounds for this order.

Nonlinear resolvents: distortion and order of starlikeness

TL;DR

The paper addresses the geometric structure of nonlinear resolvents of holomorphically accretive mappings on the open unit ball of a complex Banach space. It develops a distortion theorem to bound resolvent norms, proves that nonlinear resolvents are holomorphically accretive with a quantified squeezing ratio, and establishes that resolvents are starlike of at least order with sharper, parameter-dependent refinements. The results rely on representing generators via , employing the Riesz–Herglotz framework, and analyzing the resolvent equation to obtain precise estimates. Collectively, these findings extend one- and finite-dimensional resolvent theory to the Banach-space setting, yielding concrete bounds and conditions for accretivity and starlikeness that illuminate the dynamics of the associated holomorphic semigroups and their geometric behavior.

Abstract

This paper presents a new approach to studying nonlinear resolvents of holomorphically accretive mappings on the open unit ball of a complex Banach space. We establish a distortion theorem and apply it to address problems in geometric function theory concerning the class of resolvents. Specifically, we prove the accretivity of resolvents and provide estimates for the squeezing ratio. Further, we show that nonlinear resolvents are starlike mappings of certain order and determine lower bounds for this order.

Paper Structure

This paper contains 9 sections, 16 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. State of the art and main questions
  3. Main results and outlines
  4. Preliminary and auxiliary results
  5. Holomorphically accretive mappings
  6. Starlike mappings
  7. The class $\widehat{{\operatorname{Hol}\,}}({\mathbb B},X)$
  8. Distortion and accretivity of resolvents
  9. Order of starlikeness of resolvents

Key Result

Theorem 1.1

Let $f$ have the form $f(x)=p(x)x$, where $p\in{\operatorname{Hol}\,}({\mathbb B},{\mathbb C})$ with ${\operatorname{Re}\,} p(x) \ge a,$$x\in{\mathbb B}$, for some $a\ge0$. Let $\{G_\lambda\}_{\lambda>0}$ be the resolvent family for $f$. Denote $q=p(0)$. Then for any $\lambda>0$ we have where $A:=\left| 1 - \lambda q \right|^2+4\lambda a+1$ and $B:=\left(|1-\lambda q|^2-1\right)^2 +8\lambda^3a|q

Figures (2)

  • Figure 1: Function $\lambda\mapsto\sqrt{\frac{2}{A+\sqrt{B}}}$
  • Figure 2: Range of the parameters $s$ and $t$

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 3.1
  • Proposition 3.2
  • ...and 18 more