Numerical Range of Holomorphic Mappings and Applications

Abstract

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A sharpened form of the inverse function theorem

“…It can be easily seen that if h(0) = 0, then h is holomorphically accretive if and only if its numerical range N R(h) lies in the right half-plane. More detailed study of the numerical range of holomorphic mappings and its applications to different problems of analysis can be found in the recent book [3]. For our purpose we need the following fact.…”

The Fekete--Szegö problem and filtration of generators

“…In this case, the unique solution of (1.3) forms a semigroup of holomorphic self-mappings of the open unit disk D generated by f . Various properties of generators and semigroups generated by them can be found, for example, in recent books [3,10,11]. For our purposes we need the following characterization of generators.…”

“…The condition Re f ′ (z) ≥ 0, z ∈ D, for f ∈ A implies by the Noshiro-Warschawski Theorem (see, for example, [10,26]) that f is univalent. Since not all infinitesimal generators are univalent, the condition Re f ′ (z) ≥ 0, z ∈ D, is far from being a necessary condition for membership in G. Therefore, the following question is important.…”

Continuous and holomorphic semicocycles in Banach spaces

“…In this paper we study semicocycles over semigroups of holomorphic selfmappings of a domain in a complex Banach space (the books [19,11] and references therein can be used as good sources for the state of the art on semigroups of holomorphic mappings). While, under an assumption of uniform joint continuity (see [9]) such semicocycles can be defined as solutions of certain nonautonomous differential equations which cannot be explicitly solved, if a semicocycle is cohomolgous to a constant one it has an explicit representation.…”