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Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications
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.Comment: version 2: corrected misprints and simplified proofs in section
Linearization models for parabolic dynamical systems via Abel's functional equation
We study linearization models for continuous one-parameter semigroups of
parabolic type. In particular, we introduce new limit schemes to obtain
solutions of Abel's functional equation and to study asymptotic behavior of
such semigroups. The crucial point is that these solutions are univalent
functions convex in one direction. In a parallel direction, we find analytic
conditions which determine certain geometric properties of those functions,
such as the location of their images in either a half-plane or a strip, and
their containing either a half-plane or a strip. In the context of semigroup
theory these geometric questions may be interpreted as follows: is a given
one-parameter continuous semigroup either an outer or an inner conjugate of a
group of automorphisms? In other words, the problem is finding a fractional
linear model of the semigroup which is defined by a group of automorphisms of
the open unit disk. Our results enable us to establish some new important
analytic and geometric characteristics of the asymptotic behavior of
one-parameter continuous semigroups of holomorphic mappings, as well as to
study the problem of existence of a backward flow invariant domain and its
geometry
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