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

Abstract

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