The Schwarz-Pick lemma for slice regular functions

TL;DR

This paper extends the Schwarz-Pick lemma to quaternionic slice regular functions on the unit ball, providing new tools for quaternionic hyperbolic geometry and fixed-point analysis.

Contribution

It introduces a Schwarz-Pick lemma for slice regular functions on the quaternionic unit ball, generalizing classical complex results to quaternionic analysis.

Findings

Proves a Schwarz-Pick lemma for slice regular functions on quaternionic balls

Demonstrates applications in fixed-point theory for these functions

Generalizes the lemma to higher order derivatives

Abstract

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.