Necessary and sufficient conditions for zero subsets of holomorphic functions

TL;DR

This paper establishes conditions relating the distribution of zeros of holomorphic functions to the Riesz charge of a delta-subharmonic function that bounds the function's magnitude, extending previous results to more general functions and domains.

Contribution

It generalizes zero distribution conditions for holomorphic functions bounded by delta-subharmonic functions, expanding prior work from subharmonic functions to broader classes.

Findings

Derived zero distribution criteria using Riesz charge of delta-subharmonic functions.

Extended previous results from subharmonic to delta-subharmonic functions.

Analyzed the special case of the complex plane separately.

Abstract

Let $D$ be a domain in the complex plane, $M$ be an extended real function on $D$. If $f$ is a non-zero holomorphic function on $D$ with an upper constraint $|f|\leq \exp M$ on this domain $D$, then it is natural to expect that there must be some upper constraints on the distribution of zeros of this holomorphic function exclusively in terms of the function $M$ and the geometry of the domain $D$. We have investigated this question in detail in our previous works in the case when $M$ is a subharmonic function and the domain $D$ is arbitrary or with a non-polar boundary. The answer was given in terms of limiting the distribution of zeros of $f$ from above via the Riesz measure of the subharmonic function $M$. In this article, the function $M$ is the difference of subharmonic functions, or a $\delta$-subharmonic function, and the upper constraints are given in terms of the Riesz charge of this $\delta$-subharmonic function $M$. These results are also new to a certain extent for the subharmonic function $M$. The case when the domain D is the complex plane is considered separately. For the complex plane, it is possible to reach the criterion level.