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 ℍ in 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 Δ(0,ρ) and such that the inclusion f(Δ(0,ρ))⊇Δ(0,ρ2) 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.
