arXiv subjects
Publications and source records attributed to Chiara Della Rocchetta.
In this paper we prove the Bohr Theorem for slice regular functions. Following the historical path that led to the proof of the classical Bohr Theorem, we also extend the Borel-Carathéodory Theorem to the new setting.
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 $f'(0)=1$) 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.