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 $\mathbb{H}$ during the last decade, some natural questions arose about slice regular functions on the open unit ball $\mathbb{B}$ in $\mathbb{H}$. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of $\mathbb{B}$ fixing the origin, it establishes two variants of the quaternionic Schwarz-Pick lemma, specialized to maps $\mathbb{B}\to\mathbb{B}$ 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 $\rho$ such that $f$ is injective at least in the disk $\Delta(0,\rho)$ and such that the inclusion $f(\Delta(0,\rho))\supseteq\Delta(0,\rho^2)$ holds. The analogous result proven here for slice regular functions $\mathbb{B}\to\mathbb{B}$ allows a new approach to the study of Bloch-Landau-type properties of slice regular functions $\mathbb{B}\to\mathbb{H}$.