  We study classes of locally biholomorphic mappings defined in the $\P$ that
have bounded Schwarzian operator in the Bergman metric. We establish important
properties of specific solutions of the associated system of differential
equations and show a geometric connection between the order of the classes and
a covering property. We show for modified and slightly larger classes that the
order is Lipschitz continuous with respect to the bound on the Schwarzian, and
use this to estimate the order of the original classes.
