The theory of sharp first-order univalence criteria

A condition of the formf′ (D) ⊂R is calleda first-order univalence criterion if all functions for which it holds are univalent inD. By means of a careful analysis of the behavior of the corresponding extremal functions, we establish fundamental principles for a coherent theory of sharp criteria of this kind. In particular, we examine the relationship between three different concepts of sharpness and show how large classes of first-order univalence criteria which are sharp in a strong sense may be described in terms of the univalence properties of families of analytic functions depending on a finite number of parameters. This research was supported in part by the Chilean government through FONDECYT grant No. 0809-91.