Narrow Results

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 $𝔹\to 𝔹$ 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 $a:=|f^{\prime }(0)|$, a radius $\rho$ such that $f$ is injective at least in the disk $\mathrm{\Delta }(0,\rho )$ and such that the inclusion $f(\mathrm{\Delta }(0,\rho ))\supseteq \mathrm{\Delta }(0,{\rho }^2)$ holds. The analogous result proven here for slice regular functions $𝔹\to 𝔹$ allows a new approach to the study of Bloch–Landau-type properties of slice regular functions $𝔹\to ℍ$.