Characteristic properties of Nehari functions

Let N be the set of all meromorphic functions f defined in the unit disc D that satisfy Nehari’s univalence criterion (1 − |z|2)2|Sf(z) | ≤ 2. In this paper we investigate certain properties of the class N. We obtain sharp estimates for the spherical distortion, and also a two-point distortion theorem that actually characterizes the set N. Finally, we study some aspects of the boundary behavior of Nehari functions, and obtain results that indicate how such maps can fail to map D onto a quasidisc. 1. Introduction. Let f be analytic in the unit disc D and let Sf = (f ′′/f ′) ′ − (1/2)(f ′′/f ′)2 be its Schwarzian derivative. In 1949 Nehari showed that if (1.1) |Sf(z) | ≤ 2 (1 − |z|2)