Functions in Bloch-type spaces and their moduli

TL;DR

This paper characterizes functions in Bloch-type spaces on the unit ball of complex n-space using their moduli, providing new insights into their structure and growth conditions.

Contribution

It introduces a novel characterization of Bloch-type space functions based on their moduli, extending understanding of their regularity and growth behavior.

Findings

New modulus-based characterization of $eta_ ext{omega}$ functions

Enhanced understanding of growth conditions for holomorphic functions

Applicable to functions on the unit ball in complex n-space

Abstract

Given a suitably regular nonnegative function $\omega$ on $(0,1]$, let $\mathcal B_\omega$ denote the space of all holomorphic functions $f$ on the unit ball $\mathbb B_n$ of $\mathbb C^n$ that satisfy $$|\nabla f(z)|\le C\frac{\omega(1-|z|)}{1-|z|},\qquad z\in\mathbb B_n,$$ with some fixed $C=C_f>0$. We obtain a new characterization of $\mathcal B_\omega$ functions in terms of their moduli.