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Geometric properties of the nonlinear resolvent of holomorphic generators. (arXiv:1901.02142v1 [math.CV])
来源于:arXiv
Let $f$ be the infinitesimal generator of a one-parameter semigroup $\left\{
F_{t}\right\} _{t\ge0}$ of holomorphic self-mappings of the open unit disk
$\Delta$. In this paper we study properties of the family $R$ of resolvents
$(I+rf)^{-1}:\Delta\to\Delta~ (r\ge0)$ in the spirit of geometric function
theory. We discovered, in particular, that $R$ forms an inverse L\"owner chain
of hyperbolically convex functions. Moreover, each element of $R$ satisfies the
Noshiro-Warschawski condition and is a starlike function of order at least
$\frac12$,. This, in turn, implies that each element of $R$ is also a
holomorphic generator. We mention also quasiconformal extension of an element
of $R.$ Finally we study the existence of repelling fixed points of this
family. 查看全文>>