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

References in corpus (3)

• Slice regular functions on real alternative algebras
• Power series and analyticity over the quaternions
• Bloch's Theorem in the Context of Quaternion Analysis

Cited by in corpus (10)

• Landau-Toeplitz theorems for slice regular functions over quaternions
• Twistor interpretation of slice regular functions
• Landau's theorem for slice regular functions on the quaternionic unit ball
• Slice starlike functions over quaternions
• Quaternionic Hankel operators and approximation by slice regular functions
• On two Bloch type theorems for quaternionic slice regular functions
• Boundary Julia theory for slice regular functions
• Ideals of regular functions of a quaternionic variable
• Julia theory for slice regular functions
• On Some Geometric Properties of Slice Regular Functions of a Quaternion Variable