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

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 $f\ne 0$ satisfying the estimate $\log |f|\le {\mathsf{B}}_u$ on $\mathbb{C}$, where functions ${\mathsf{B}}_u$ are integral averages of $u$ for rapidly shrinking disks as it approaches infinity. We give another equivalent version of this result with $\log |f|\le u$ outside a very small exceptional set if $u$ is of finite order.