Approximation of subharmonic functions of slow growth

Анотація

Let $u$ be a subharmonic function in $\mathbb{C}$, $\mu_u$ its Riesz measure. Suppose that $C_1\le \mu(\{z: R<|z|\le R\psi(R)\})\le C_2 $ $ (R\ge R_1)$ for some positive constants $C_1$, $C_2$, and $R_1$, and a slowly growing to $+\infty$ function $\psi(r)$ such that $r/\psi(r)\nearrow +\infty$ ($r\to\infty$). Then there exist an entire function $f$, constants $K_1=K_1(C_1, C_2),$ $K_2=K_2(C_2)$ and a set $E\subset \mathbb{C}$ such that
$$|u(z)-\log|f(z)||\le K_1 \log \psi(|z|), \; z\to\infty, z \not\in E,$$ and $E$ can be covered by the system of discs $D_{z_k}(\rho_k)$ satisfying $$ \displaystyle\sum_{R<|z_k|<R\psi(R)}\frac{\rho_k\psi(|z_k|)}{|z_k|}<K_2,$$ as $R_2\to+\infty$. We prove also that the estimate of the exceptional set is sharp up to a constant factor.