The restriction from below of the subharmonic function by the logarithm of the module of entire function

Abstract

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 $r$.