The Logarithm of the Modulus of an Entire Function as a Minorant for a Subharmonic Function outside a Small Exceptional Set

TL;DR

This paper establishes a new version of a result on constructing entire functions that minorize subharmonic functions outside a small exceptional set, especially for functions of finite order, refining previous integral average estimates.

Contribution

It provides an alternative formulation of existing results, showing the existence of entire functions bounded by the subharmonic function outside a negligible set for finite order functions.

Findings

Existence of entire functions bounded by subharmonic functions outside small sets

Equivalent formulation using pointwise bounds instead of integral averages

Applicable to subharmonic functions of finite order

Abstract

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. In 2016, we obtained a result on the existence of an entire function $f\neq 0$ satisfying the estimate $\log|f|\leq {\sf B}_u$ on $\mathbb C$, where functions ${\sf B}_u$ are integral averages of $u$ for rapidly shrinking disks as it approaches infinity. We give another equivalent version of this result with $\log |f|\leq u$ outside a very small exceptional set if $u$ is of finite order.