The Schwarzian Derivative for Convex Holomorphic Mappings in Several Complex Variables

We obtain upper bounds for the norm of the Schwarzian derivative of convex holomorphic mappings defined on the polydisk and the unit ball in ${\mathbb{C}}^n$. For coordinate-wise convex mappings on the polydisk, we derive a sharp estimate extending the classical one-variable result of Chuaqui–Duren–Osgood to higher dimensions. For the Roper–Suffridge extension operator in the unit ball, we obtain an explicit bound that represents the best available estimate in this setting.