The growth of subharmonic functions along the imaginary axis

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.