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

TL;DR

This paper establishes a method to construct entire functions that bound subharmonic functions from below using their logarithmic modules, with applications to functions of finite order and small exceptional sets.

Contribution

It introduces new bounds for subharmonic functions via entire functions, extending classical results to variable radii and small exceptional sets.

Findings

Existence of entire functions bounding subharmonic functions with variable radius

Extension of bounds to subharmonic functions of finite order

Control of exceptional sets using Hausdorff content

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 $r$.