Behavior of Subharmonic Functions of Slow Growth Outside Exclusive Sets

Let v be a function of slow growth unbounded on [0, + ∞), let u be subharmonic (in plane) function of order zero, let μ be its Riesz measure, and let n(t, u) = μ({x : |x| ≤ t}), \(N\left(t,u\right)={\int }_{1}^{t}n\left(\tau ,u\right)/\tau d\tau ,\) $N\left(t,,,u\right)={\int }_1^{t}n\left(\tau ,,,u\right)/\tau d\tau ,$ and n(r, u) = O(v(r)), r → + ∞. A set E ∈ ℂ is called a \({C}_{0}^{\beta }\) $C_0^{\beta }$ -set, 0 < β ≤ 1, if E can be covered with a system of disks K(a_n, r_n) = {z : |z − a_n| < r_n} such that \({\sum }_{\left|{a}_{n}\right|\le r}{r}_{n}^{\beta }=o\left({r}^{\beta }\right),r\to +\infty .\) ${\sum }_{\left|a_n\right|\le r}{r}_n^{\beta }=o\left(r^{\beta }\right),r\to +\infty .$

Then, for every nondecreasing function ϕ unbounded on [0, +1), there exists a \({C}_{0}^{\beta }\) $C_0^{\beta }$ -set E such that

\(u\left(z\right)=N\left(r,u\right)+o\left(\phi \left(r\right)v\left(r\right)\right), r=\left|z\right|\to +\infty , z\notin E.\) $u\left(z\right)=N\left(r,,,u\right)+o\left(\varphi ,\left(r\right),v,\left(r\right)\right),r=\left|z\right|\to +\infty ,z\notin E.$

It is shown that, in this asymptotic formula, the remainder o(φ(r)v(r)) cannot be replaced by O(v(r)).