Geometric properties of the nonlinear resolvent of holomorphic generators

Let $f$ be the infinitesimal generator of a one-parameter semigroup $left{ F_{t}right} _{tge0}$ of holomorphic self-mappings of the open unit disk $Delta$. In this paper we study properties of the family $R$ of resolvents $(I+rf)^{-1}:DeltatoDelta~ (rge0)$ in the spirit of geometric function theory. We discovered, in particular, that $R$ forms an inverse Lowner 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.