A UNIFIED APPROACH TO UNIVALENCE CRITERIA IN THE UNIT DISC

From the recent injectivity criterion of Osgood and Stowe we recover m any of the known univalence criteria in the unit disc D and derive as well new conditions on D and simply-connected domains. While the crite ria of Epstein can be established in this fashion, we show how the 'di ameter term' in the criterion of Osgood and Stowe gives a sharper form of a condition of Ahlfors. Finally, on simply-connected domains we fi nd a sufficient condition for univalence that is the counterpart to a necessary one proved by Bergman and Schiffer.