Some properties of the set of schwarzians of conformal functions

Let U denote the set of all Schwarzian derivatives $S_f$ of conformal function f in the unit disk D. We show that if $S_f$ is a local extreme point of U, then f cannot omit an open set. We also show that if $S_f \in U$ is an extreme point of the closed convex hull $\bar{co}U$ of U, then f cannot omit a set of positive area. The proof of this uses Nguyen's theorem.