The growth of subharmonic functions along the imaginary axis

TL;DR

This paper investigates the dominance relationship between subharmonic functions in the complex plane based on their growth along a line, providing quantitative measures and applications to uniqueness theorems for entire functions of exponential type.

Contribution

It introduces quantitative criteria for measure dominance between subharmonic functions based on their growth, with applications to uniqueness theorems for entire functions.

Findings

Established conditions for measure dominance based on growth comparisons.

Provided a new uniqueness theorem for entire functions of exponential type.

Demonstrated the results with explicit examples and applications.

Abstract

Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ are two subharmonic functions in the complex plane $\mathbb C$ with the Riesz measures $\nu_u$ and $\mu_M$ such that $u(z)\leq O(|z|)$ and $M(z)\leq O(|z|)$ as $z\to \infty$. If the growth of a function $M$ in some sense exceeds the growth of a function $u$ on some straight line, then we can expect measure $\mu_M$ to dominate measure $\nu_u$ in some sense. We give quantitative forms of such dominance. The main results are illustrated by a new uniqueness theorem for entire functions of exponential type.