General Univalence Criteria in the Disk: Extensions and Extremal Function

Many classical univalence criteria depending on the Schwarzian derivative are special cases of a result, proved in [18], involving both conformal mappings and conformal metrics. The classical theorems for analytic functions on the disk emerge by choosing appropriate conformal metrics and computing a generalized Schwarzian. The results in this paper address questions of extending functions which satisfy the general univalence criterion; continuous extensions to the closure of the disk, and homeomorphic and quasiconformal extensions to the sphere. The main tool is the convexity of an associated function along geodesics of the metric. The other important aspect of this study is an extremal function associated with a given criterion, along with its associated extremal geodesics. An extremal function for a criterion is one whose image is not a Jordan domain. An extremal geodesic joins points on the boundary which map to the same point in the image. We show that, for the general criterion, the image of an extremal geodesic under an extremal function is a euclidean circle.