On convex mappings

TL;DR

This paper characterizes convex conformal mappings of the unit disk, identifies extremal mappings onto half-planes or strips, and shows that level sets of the Poincaré metric are strictly convex except in specific cases.

Contribution

It provides a new characterization of convex conformal mappings and identifies extremal cases, advancing understanding of geometric properties of such mappings.

Findings

Characterization of convex conformal mappings of the unit disk.

Identification of extremal mappings onto half-planes or strips.

Proof that level sets of the Poincaré metric are strictly convex except in special cases.

Abstract

We establish a new characterization for a conformal mapping of the unit disk $\mathbb{D}$ to be convex, and identify the mappings onto a half-plane or a parallel strip as extremals. We also show that, with these exceptions, the level sets of $\lambda$ of the Poincar\'e metric $\lambda|dw|$ of a convex domain are strictly convex.