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

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.