A Bloch-Landau Theorem for slice regular functions

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 D under a holomorphic function f (such that f(0)=0 and f'(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 H of quaternions. If f is a regular function on the open unit ball B⊂H, then for every w in B we define the regular translation 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 B through some regular translation f_w of any slice regular function f:B→H (such that f(0)=0 and ∂Cf(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.'