On geometric aspects of quaternionic and octonionic slice regular functions

TL;DR

This paper advances the geometric understanding of quaternionic and octonionic slice regular functions by establishing boundary Schwarz lemmas, Landau-Toeplitz theorems, and other fundamental results, with new methods and improved estimates.

Contribution

It introduces new geometric results and methods for octonionic and quaternionic slice regular functions, including boundary Schwarz lemmas and extremal function characterizations.

Findings

Proved a boundary Schwarz lemma for octonionic slice regular self-mappings.

Derived Landau-Toeplitz type theorems for slice regular functions.

Improved the boundary Schwarz lemma for quaternionic slice regular functions with optimal estimates.

Abstract

The purpose of this paper is twofold. One is to enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau-Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow to prove the minimum principle and one version of the open mapping theorem. Another is to strengthen a version of boundary Schwarz lemma first proved in \cite{WR} for quaternionic slice regular functions, with a completely new approach. Our quaternionic boundary Schwarz lemma with optimal estimate improves considerably a well-known Osserman type estimate and provides additionally all the extremal functions.