A Bloch-Landau Theorem for slice regular functions

Abstract

The Bloch-Landau Theorem is one of the basic results in the geometric theory of holomorphic functions. It establishes that the image of the open unit disc $\mathbb{D}$ under a holomorphic function $f$ (such that $f(0)=0$ and $f^{\prime }(0)=1$) always contains an open disc with radius larger than a universal constant. In this paper we prove a Bloch-Landau type Theorem for slice regular functions over the skew field $\mathbb{H}$ of quaternions. If $f$ is a regular function on the open unit ball $\mathbb{B}\subset \mathbb{H}$, then for every $w\in \mathbb{B}$ we define the regular translation ${\tilde{f}}_w$ of $f$. The peculiarities of the non commutative setting lead to the following statement: there exists a universal open set contained in the image of $\mathbb{B}$ through some regular translation ${\tilde{f}}_w$ of any slice regular function $f:\mathbb{B}\to \mathbb{H}$ (such that $f(0)=0$ and ${\partial }_{C}f(0)=1$). For technical reasons, we introduce a new norm on the space of regular functions on open balls centred at the origin, equivalent to the uniform norm, and we investigate its properties.

17 pages

Cited by in corpus (2)

• Landau-Toeplitz theorems for slice regular functions over quaternions
• On Some Geometric Properties of Slice Regular Functions of a Quaternion Variable