Necessary and Sufficient Conditions for Zero Subsets of Holomorphic Functions with Upper Constraints in Planar Domains

Abstract

Let $D$ be a domain in the complex plane and $M$ be an extended real-valued function on $D$. If $f$ is a non-zero holomorphic function on $D$ such that $|f|\leq\exp M$, then it is natural to expect that there should be some upper boundedness for the distribution of the zeros of $f$ expressed 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 $D$ either is arbitrary or has a non-polar boundary. The answer was given in terms of constraints to 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 $D$ coincides with the whole complex plane is considered separately. For the complex plane, it is possible to reach the criterion level of our results.