# Free Holomorphic Functions on Polydomains

## TL;DR

This paper extends the theory of free holomorphic functions on noncommutative polydomains by establishing classical complex analysis results and properties in a multivariable noncommutative setting.

## Contribution

It introduces analogues of classical theorems for free holomorphic functions on noncommutative polydomains, advancing the understanding of their structure and properties.

## Key findings

- Established Abel theorem, Hadamard formula, Cauchy inequality, and Liouville theorem in the noncommutative multivariable setting.
- Proved maximum principle and Schwarz type lemma for free holomorphic functions.
- Showed the algebra of free holomorphic functions forms a complete metric space.

## Abstract

In this paper, we continue to develop the theory of free holomorphic functions on noncommutative regular polydomains. We find analogues of several classical results from complex analysis such as Abel theorem, Hadamard formula, Cauchy inequality, and Liouville theorem for entire functions, in our multivariable setting. We also provide a maximum principle and a Schwarz type lemma. These results are used to prove analogues of Weierstrass, Montel, and Vitali theorems for the algebra of free holomorphic functions on the regular polydomain, which turns out to be a complete metric space.