Behavior of subharmonic functions of slow growth outside exclusive sets

Abstract

UDC517.53

Let $v$ be a slowly growing function unbounded on $[0,\,+\infty),$ $u$ be subharmonic (in plane) function of zero order, $\mu$~be its Riesz measure, $n(t,u)=\mu(\{x\colon |x|\le t\}),$ $N(t,u)=\int_{1}^{t}n(\tau,u)/\tau d\tau,$ and $n(r,u)=O(v(r)),$ $r\to+\infty.$  A  set $E \in \mathbb{C}$ is called a $C_0^\beta$-set, $0 < \beta \le 1,$ if $E$ can be covered by a system of disks $K(a_n,r_n)=\{z\colon |z-a_n| < r_n\}$ such that $\sum_{|a_n| \le r} r_n^\beta = o(r^\beta),$ $r\to+\infty.$ Then, for every nondecreasing function  $\phi$ unbounded on $[0,\,+\infty),$  there exists a $C_0^\beta$-set $E$ such that  $$\begin{equation*}u(z)=N(r,u)+o(\phi(r)v(r)),\qquad r=|z|\to+\infty,\quad z \notin E.\end{equation*}$$  It is shown that, in this asymptotic formula, the remainder term $o(\phi(r)v(r))$ cannot be changed by $O(v(r)).$