Linear invariance and uniform local univalence

Abstract

Pommerenke initiated the study of linearly invariant families of functions defined in the unit disk D. A holomorphic function f on D is called linearly invariant if the family of all Koebe transforms of f has finite linear invariant order. A function f is linearly invariant on D if and only if f is uniformly locally univalent in the hyperbolic sense; that is, there is an r > 0 such that f is univalent in every hyperbolic disk of radius r. We present two extensions of the notion of linear invariance to general planar regions, one involves the hyperbolic metric and the other the quasihyperbolic metric. We relate these two concepts of linear invariance to uniform local univalence relative to each of these metrics. For uniformly perfect regions all of these concepts coincide; we obtain various inclusion relations for non-uniformly perfect regions. Finally, we characterize entire functions which are uniformly locally univalent relative to the euclidean metric and establish a curious connection between function...