Families of Homomorphic Mappings in the Polydisk

TL;DR

This paper investigates classes of locally biholomorphic mappings in the polydisk with bounded Schwarzian operators, exploring their properties, geometric connections, and how the order varies with bounds, contributing to complex analysis and geometric function theory.

Contribution

It introduces new classes of mappings with bounded Schwarzian in the polydisk and analyzes their properties, including the Lipschitz continuity of their order relative to bounds.

Findings

The order of the classes is Lipschitz continuous with respect to the Schwarzian bound.

Established geometric connections between the order and covering properties.

Provided estimates for the order of original classes based on modified classes.

Abstract

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.