Approximation of subharmonic functions

Given an arbitrary subharmonic function of finite order on the plane, an entire function $f(z)$ is constructed which satisfies the asymptotic relation
$$ |u(z)-\ln|f(z)||\leqslant C\ln^2|z|,\qquad|z|\to\infty, $$
outside a sufficiently small exceptional set $E$. Functions with a logarithmic estimate are constructed in some special cases.
Bibliography: 3 titles.