The growth of subharmonic functions along the imaginary axis

Let $u\equiv ̸-\mathrm{\infty }$ and $M\equiv ̸-\mathrm{\infty }$ are two subharmonic functions in the complex plane $\mathbb{C}$ with the Riesz measures ${\nu }_u$ and ${\mu }_M$ such that $u(z)\le O(\mathrm{\mid }z\mathrm{\mid })$ and $M(z)\le O(\mathrm{\mid }z\mathrm{\mid })$ as $z\to \mathrm{\infty }$. 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.