Distribution of zeros and masses for holomorphic and subharmonic functions. I. Hadamard- and Blaschke-type conditions

Abstract

Let $M$ be a subharmonic function on a domain $D$ in the complex plane $\mathbb{C}$ with the Riesz measure ${\nu }_M$. Let $f$ be a non-zero holomorphic function on $D$ such that $\log |f|\le M$ on $D$ and the function $f$ vanish on a sequence $\mathtt{Z}=\{{\mathtt{z}}_k{\}}_{k=1,2,\dots }\subset D$ {\large(}$u\not\equiv -\infty$ be a subharmonic function on $D$ with the Riesz measure or the mass distribution ${\nu }_u$, and $u\le M$ on $D$ resp.{\large)}. Then restrictions on the growth of the Riesz measure ${\nu }_M$ of the function $M$ near the boundary of the domain $D$ entail certain restrictions on the distribution of points of the sequence $\mathtt{Z}$ (to the mass distribution ${\nu }_u$ resp.). A quantitative form of research of this phenomenon is given immediately in the subharmonic framework. We also establish results in the inverse direction. We investigated in detail the cases when $D$ is $\mathbb{C}$, the unit disk, exterior of the unit disk, a concentric annulus, and $M$ is a radial function; $D$ is a regular domain and $M$ are constant on the level lines of Green's function of this domain $D$; $D$ is a domain of hyperbolic type, and $M$ are the superpositions of convex functions with functions that depend on the hyperbolic radius; $D$ is a regular domain and $M$ is the superposition of convex functions with a function dependent on the distance to some subset of the boundary of the domain $D$. All our main results and their implementation in more or less concrete situations are new not only for subharmonic functions $u$, and also for holomorphic functions $f$ even in the case when $D$ is $\mathbb{C}$, the unit disk, an annulus etc.

70 pages, list of references 81, the text of our work in Russian. The work will be published in parts in various publications

References in corpus (3)

• Uniqueness Theorems for Subharmonic and Holomorphic Functions of Several Variables on a Domain
• Holomorphic minorants of plurisubharmonic functions
• On distribution of zeros of holomorphic functions: Blaschke-type conditions