Preorders on Subharmonic Functions and Measures with Applications to the Distribution of Zeros of Holomorphic Functions

Abstract

Let $X$ be a class of extended numerical functions on a domain $D$ of $d$-dimensional Euclidean space $\mathbb R^d$, $H\subset X$. Given $u,M\in X$, we write $u\prec_H M$ if there is a function $h\in H$ such that $u+h\leq M$ on $D$. We consider this special preorder $\prec_H$ for a pair of subharmonic unctions $u, M$ on $D$ in cases where $H$ is the space of all harmonic functions on $D$ or $H$ is the convex cone of all subharmonic functions $h \not\equiv -\infty$ on $D$. Main results are dual equivalent forms for this preorder $\prec_H$ in terms of balayage processes for Riesz measures of subharmonic functions $u$ and $M$, for Jensen and Arens-Singer (representing) measures, for potentials of these measures, and for special test functions generated by subharmonic functions on complements $D\setminus S$ of non-empty precompact subsets $S\Subset D$. Applications to holomorphic functions $f$ on a domain $D\subset \mathbb C^n$ relate to the distribution of zero sets of functions $f$ under upper restrictions $|f|\leq \exp M$ on $D$. If a domain $D\subset \mathbb C$ is a finitely connected domain with non-empty exterior or a simply connected domain with two different points on the boundary of $D$, then our conditions for the distribution of zeros of $f\neq 0$ with $|f|\leq \exp M$ on $D$ are both necessary and sufficient.