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 $$u\in X$$ and $$M\in X$$ , 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 functions $$u$$ and $$M$$ on $$D$$ in cases when $$H$$ 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 $$D\backslash S$$ 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.