Regular behavior of subharmonic in space functions of the zero kind

Abstract

Let $u$ be a subharmonic in $\mathbb{R}^m$, $m\geq 3$, function of the zero kind with Riesz measure $\mu$ on negative axis $Ox_1$, $n(r,u)=\mu\left(\{x\in\mathbb{R}^m \colon |x|\leq r\}\right)$, $$N(r,u)=(m-2)\int_1^r n(t,u)/t^{m-1}dt,$$ $\rho(r)$ is a proximate order, $\rho(r)\to\rho$ as $r\to+\infty$, $0<\rho<1$. We found the asymptotic of $u(x)$ as $|x|\to+\infty$ by the condition $N(r,u)=\left(1+o(1)\right)r^{\rho(r)}$, $r\to+\infty$. We also investigated the inverse relationship between a regular growth of $u$ and a behavior of $N(r,u)$ as $r\to+\infty$.