The Schwarzian Derivative for Convex Holomorphic Mappings in Several Complex Variables

TL;DR

This paper establishes upper bounds for the Schwarzian derivative norm of convex holomorphic mappings in several complex variables, extending classical results to higher dimensions and specific operators.

Contribution

It provides new sharp estimates for the Schwarzian derivative in multiple complex variables, including the polydisk and the unit ball, improving existing bounds.

Findings

Derived a sharp estimate for coordinate-wise convex mappings on the polydisk.

Obtained an explicit, optimal bound for the Roper--Suffridge extension operator.

Extended classical one-variable results to higher-dimensional complex analysis.

Abstract

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.