  Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane
$\mathbb C$. Then for any function $r\colon\mathbb C\to (0,1]$ satisfying the
condition $$\inf_{z\in\mathbb C}\frac{\ln r(z)}{\ln(2+|z|)}>-\infty,$$ there is
an entire function $f\not\equiv 0$ such that $$ \ln |f(z)|\leq
\frac{1}{2\pi}\int_0^{2\pi}u(z+r(z)e^{i\theta})\,{\mathrm
d}\theta\quad\text{for all $z\in\mathbb C$.}$$ A similar result is established
for subharmonic functions of finite order with inequalities of the form
$\ln|f(z)|\leq u(z)$ at all points $z\in\mathbb C\setminus E$, where the
exceptional set $E$ is small in terms of $d$-dimensional Hausdorff content of
$E$ with variable radius $r$.
