On geometric aspects of quaternionic and octonionic slice regular functions (Q1747532)

Analogues of classically known geometric aspects for holomorphic functions in the unit disc are explored in a well-written and interesting manuscript. Principal results shown are for slice regular functions on domains over the octonions \({\mathbf O}.\) The first main result is a boundary Schwarz lemma for slice regular self-mappings of the open unit ball \(\mathbf B =\{ w \in\mathbf O : |w| < 1 \}\). A quaternionic boundary Schwarz lemma was obtained by \textit{G. Ren} and \textit{X. Wang} [Trans. Am. Math. Soc. 369, No. 2, 861--885 (2017; Zbl 1372.30054)] which is proved anew in the current work by a completely different approach yielding the functions for the optimal result. The latter improves on a well-known Osserman-type estimate providing all the extremal functions. An application of the theorem for octonionic slice regular functions noted above gives a Landau-Toeplitz-type theorem whose quaternionic version was proved in [\textit{G. Gentili} and \textit{G. Sarfatti}, Pac. J. Math. 265, No. 2, 381--404 (2013; Zbl 1284.30046)]. A second application of the main result noted above leads to a Cauchy-type estimate (an analogue to one of \textit{K. A. Poukka} [Arch. der Math. u. Phys. (3) 12, 251--254 (1907; JFM 38.0433.03)]) for which the explicit functions providing equality are determined. Ideas developed in the proof of the main result for octonionic slice regular functions are used to obtain on symmetric slice domains a minimum principle and an open mapping theorem. The latter is used for proving an octonionic version of the classical Koebe one-quarter theorem for slice regular extensions to the octonionic ball \(\mathbf B\) of univalent holomorphic functions on the unit disc of the complex plane.