Approximation of subharmonic functions of slow growth

Анотація

Let u be a subharmonic function in ℂ , m_u its Riesz measure. Suppose that C₁≤m({z : R < |z| ≤ Ry (R)} ≤ C₂(R ≥ R₁) for some positive constants C₁, C₂, and R₁, and a slowly growing to +¥ function y(r) such that r/y(r) ↗ +¥ (r®+¥). Then there exist an entire function f, constants K₁ = K₁(C₁; C₂), K₂ = K₂(C₂) and a set E Ì ℂ such that
|u(z)-log|f(z)||≤K₁logy(|z|), z®¥, z∉ E,
and E can be covered by the system of discs D_zk (r_k) satisfying
,
as R₂® +¥. We prove also that the estimate of the exceptional set is sharp up to a constant factor.