Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications

TL;DR

This paper provides bounds and geometric insights into the numerical range of holomorphic mappings in Banach spaces, with applications to fixed point theory, radii of starlikeness, and complex dynamical systems.

Contribution

It establishes new lower and upper bounds for the numerical range of holomorphic mappings and explores their implications in various geometric and analytic contexts.

Findings

Derived bounds for the numerical range in Banach spaces

Analyzed fixed point and resolvent properties of holomorphic mappings

Investigated radii of starlikeness and spirallikeness

Abstract

The numerical range of holomorphic mappings arises in many aspects of nonlinear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite and infinite dimensional Banach spaces, and in the study of complete and semi-complete vector fields and their applications to starlike and spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic mappings. In the present paper we establish lower and upper bounds for the numerical range of holomorphic mappings in Banach spaces. In addition, we study and discuss some geometric and quantitative analytic aspects of fixed point theory, nonlinear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness.