A characterization of holomorphic mappings on a poly-plane

Abstract

We show that any function $f:{\mathbb{H}}^n\to \mathbb{H}$ with $f(z+c)=f(z)+c$, $z\in {\mathbb{H}}^n$, for some $c>0$ has a property that any limit function of a family $\{\frac{f(tz)}{t}{\}}_{t>0}$ when $t\to \infty$ is linear.