Some Criteria of Univalence1

is the Schwarzian derivative of w =/(z). The two cases treated in [3 ] were m(\z\) =ir2/2 and m(\z\) =2(1 — \z\ 2)~2. The constants appearing in both criteria are the largest possible. In the first case this is shown by the existence of the nonunivalent function w = tan ir( 1 + e)z/2 (e>0) for which {w, z\ =7r2(l+e)2/2, and in the second case by an example constructed by E. Hille [2]. Other criteria of this type have meanwhile been announced (without proof) by V. Pokornyi [4], the only sharp one among them being the one corresponding to w(|z|) = 4(1 | z\2)-1, with the extremal /(z) =/^(l -z2)~2dz. The main objective of the present note is to establish the following more general criterion of univalence.