A non-commutative Julia Inequality

TL;DR

This paper establishes a Julia inequality for non-commutative functions on polynomial polyhedra and applies it to classical domains, providing insights into boundary differentiability under regularity conditions.

Contribution

It introduces a Julia inequality in the non-commutative setting and extends it to classical complex domains, linking non-commutative and commutative function theory.

Findings

Proves a Julia inequality for non-commutative functions

Derives a Julia inequality for holomorphic functions in complex domains

Analyzes boundary differentiability with regularity assumptions

Abstract

We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in $\mathbb{C}^d$. We look at differentiability at a boundary point for functions that have a certain regularity there.