Landau’s theorem for slice regular functions on the quaternionic unit ball

Abstract

Along with the development of the theory of slice regular functions over the real algebra of quaternions H during the last decade, some natural questions arose about slice regular functions on the open unit ball B in H. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of B fixing the origin, it establishes two variants of the quaternionic Schwarz-Pick lemma, specialized to maps B → B that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps f of the complex unit disk with f (0) = 0. Landau had computed, in terms of a := |f ′ (0)|, a radius ρ such that f is injective at least in the disk ∆(0, ρ) and such that the inclusion f (∆(0, ρ)) ⊇ ∆(0, ρ 2 ) holds. The analogous result proven here for slice regular functions B → B allows a new approach to the study of Bloch-Landau-type properties of slice regular functions B → H.

On a quaternionic Picard theorem

“…This implies that η extends to a holomorphic map defined on Δ (Proposition 3.2). Since over the complex field, Picard's theorems are the global version of the local Landau's Theorem, we point out that a quaternionic Landau's Theorem for slice regular functions already exists in the literature; see [3]. Proof.…”

“…Then the following are equivalent: 3 i=0 v 2 i = 0. (iii) There exists an imaginary unit H ∈ S such that Hv = v .…”

On quaternionic tori and their moduli space

“…To conclude the Introduction, we point out that the theory of slice regular functions, presented in detail in the monograph [18], has been applied to the study of a non-commutative functional calculus, (see for example the monograph [9] and references therein) and to address the problem of the construction and classification of orthogonal complex structures in open subsets of the space H of quaternions (see [17]). Recent results of geometric theory of regular functions appear in [3], [4], [5], [6], [7], [12], [13], [21], [22], [23]. Paper [14] is strictly related to the topic of the present article.…”

Slice-Quaternionic Hopf Surfaces

“…Such a notion has its source in the work by Fueter, further developed by Ghiloni and Perotti. It represents a counterpart in several variables of the notion of slice-regular function in one quaternionic variable studied in [GS06,GS07], which appeared to share with holomorphic functions a rich theory from the analytic point of view, [BS12], [BS17]; see also [CSS12]. We refer to [GSS13,GP12] for precise definitions and for results.…”