On Geometric Aspects of Quaternionic and Octonionic Slice Regular Functions

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 [37] 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.2010 Mathematics Subject Classification. Primary 30G35; Secondary 30C80, 32A40, 31B25.

Growth and distortion theorems for slice monogenic functions

“…The counterpart of the convex combination identity (3.11) in Lemma 3.2 also holds for slice regular functions defined on octonions or more general real alternative algebras under the extra assumption that f preserves at least one slice. This can be verified similarly as in the proof of Proposition 3.1, see [29,32] for details.…”

“…Furthermore, only in the quaternionic setting, we have an explicit formula to express of the regular product and regular quotient in terms of the usual pointwise product and quotient. It is exactly this explicit formula which plays a crucial role in many arguments, see the monograph [16] and the recent papers [28,32] for more details. In higher dimensions, the formulas to express slice product and slice quotient in terms of the usual pointwise ones hold true only under some special cases, see [22, Corollary 3.5 and Theorem 3.7] for details.…”

Division algebras of slice functions

“…The latter principle is, in turn, applied to prove the open mapping theorem. These results subsume the separate results of [1,7,9,11,23] and they strengthen them both over quaternions and over octonions.…”

“…The previous example shows that [23,Formula (5.2)] is only true under additional assumptions, such as those of Remark 4.6. We are now ready to provide an example where T f admits an extension to Ω ′ , though not through formula (12), and an example where it does not.…”

“…Quaternionic slice regular functions have been extensively studied: see [13] for a survey of the first phase of their study. Over the octonions, power series have been investigated in [16], while slice regular functions have been considered in the recent work [23]. A key tool for these studies was a quaternionic result called representation formula, [4, theorem 2.27], along with the related octonionic result [16, lemma 1].…”