Lipschitz Estimates for Conformal Maps from the Unit Disk to Convex Domains

TL;DR

This paper provides an explicit upper bound for the derivative of conformal maps from the unit disk to convex domains, depending on geometric boundary parameters, using a hyperbolic-type metric.

Contribution

It introduces a new uniform derivative estimate for conformal maps onto convex domains based on boundary radii and curvature, utilizing a M"obius invariant hyperbolic metric.

Findings

Derived an explicit upper bound for conformal map derivatives

Bound depends only on domain radii and boundary curvature

Utilized a M"obius invariant hyperbolic metric in the proof

Abstract

We obtain an explicit uniform upper bound for the derivative of a conformal mapping of the unit disk onto a convex domain. This estimate depends only on the outer and inner radii of the domain, and on a curvature radius of its boundary. Its proof is based on a M\"obius invariant metric of hyperbolic type, introduced by Kulkarni and Pinkall in 1994.