Schwarz Lemma for the tetrablock

TL;DR

This paper characterizes all complex geodesics in the tetrablock passing through the origin, providing a detailed description of extremals in the Schwarz Lemma for this domain, and extends previous results on the topic.

Contribution

It explicitly describes all complex geodesics in the tetrablock through the origin, advancing the understanding of extremals in the Schwarz Lemma for this domain.

Findings

All complex geodesics passing through the origin are characterized.

Descriptions of extremals for the Lempert function are provided.

A general form of complex geodesics in domains is established.

Abstract

We describe all complex geodesics in the tetrablock passing through the origin thus obtaining the form of all extremals in the Schwarz Lemma for the tetrablock. Some other extremals for the Lempert function and geodesics are also given. The paper may be seen as a continuation of the results Abouhajar, White and Young. The proofs rely on a necessary form of complex geodesics in general domains which is also proven in the paper.