Schwarzian derivatives and some criteria for univalence in

Oda gave a definition for Schwarzian derivatives in several complex variables [Oda, T., 1974, On Schwarzian derivatives in several variables (in Japanese). Kokyuroku Research Institute for Mathematical Sciences, Kioto University, 226, 82–85.], which we used in [Hernández, R., Schwarzian derivatives and linear invariant families in (Formula presented.). Journal of Mathematics (to appear).] to define a Schwarzian operator. In this article, we prove that different kinds of bounds on the norm of the Schwarzian operator imply univalence in the unit ball (Formula presented.) and on convex domains. We adapt Sturm comparison techniques to study the zero sets of a set of linearly independent solutions to a system of differential equations associated with the Schwarzian derivatives.