Moduli of holomorphic functions and logarithmically convex radial weights

TL;DR

This paper characterizes radial weights on the unit disk for which holomorphic functions can approximate these weights, extending results to several complex variables and convex domains.

Contribution

It provides a characterization of radial weights with holomorphic functions approximating them, including extensions to higher-dimensional convex domains.

Findings

Identifies conditions for weights to be equivalent to sums of holomorphic functions

Extends results from the unit disk to several complex variables

Provides a framework for understanding weights in complex analysis

Abstract

Let $H(D)$ denote the space of holomorphic functions on the unit disk $D$. We characterize those radial weights $w$ on $D$, for which there exist functions $f, g \in H(D)$ such that the sum $|f| + |g|$ is equivalent to $w$. Also, we obtain similar results in several complex variables for circular, strictly convex domains with smooth boundary.