A characterization of holomorphic mappings on a poly-plane

TL;DR

This paper investigates the asymptotic behavior of holomorphic functions on poly-planes that satisfy a specific translation property, showing that their scaled limits are linear functions.

Contribution

It characterizes the limit functions of a class of holomorphic mappings on poly-planes with a translation invariance property.

Findings

Limit functions of scaled holomorphic maps are linear.

Functions with the given translation property exhibit linear asymptotic behavior.

The result extends understanding of the structure of holomorphic functions on poly-planes.

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.