Rigidity for regular functions over Hamilton and Cayley numbers and a boundary Schwarz' Lemma

The Schwarz-Pick lemma for slice regular functions

“…As an application of the main theorem, in section 4 we obtain direct proofs of the quaternionic analogs of the Cartan Rigidity theorems mentioned at the beginning of this introduction. Versions of these results have been proven in [13], and our new approach allows to strengthen their statements. (1) f coincides with the identity function;…”

“…As an application of our main result 3.7, we study the fixed point case extending the work done in [13]. In the proofs, we thoroughly use the following property of the zero set proven in [9] (an immediate consequence of theorem 2.5).…”

Recent Developments for Regular Functions of a Hypercomplex Variable

“…Nevertheless, since the regularity of functions is not preserved under composition, the above mentioned Cayley transformation and regular Moebius transformations do not help in generalizing theorem 6.1. The following is indeed proven by using a different technique (see [21]):…”

“…In [38] the authors use a more direct approach to prove a generalization of Theorem 6.6 which gives some insight for other rigidity results. For the regular case in [21] the following analogue of Theorem 6.6 is obtained Theorem 6.7. Let f be a regular self-map of B(0, 1) ⊂ K. Assume there exists…”

“…Before focussing our attention on other peculiar properties of regular functions, we want to devote this section to some rigidity properties shared by regular and holomorphic functions (see -for the holomorphic case -, e.g., [2,26,31,38]; for the regular case we refer to [21]). We begin by stating an analogue of the classical Schwarz lemma for regular functions: its proof (see [18,19]) is not a consequence of the Splitting Lemmas.…”