Accuracy of approximation of subharmonic functions by logarithms of moduli of analytic ones in Chebyshev metrics

TL;DR

This paper investigates how the precision of approximating subharmonic functions by logarithms of entire functions affects the size of the exceptional set, revealing a trade-off between accuracy and exceptional set size.

Contribution

It establishes new bounds showing that increasing approximation accuracy enlarges the exceptional set, extending previous results to functions of infinite order and in the unit disk.

Findings

Decreasing the approximation constant C increases the exceptional set size.

Results apply to subharmonic functions of both finite and infinite order.

Improves and complements Yulmukhametov's earlier work.

Abstract

It is known that a subharmonic function of finite order $\rho$ can be approximated by the logarithm of the modulus of an entire function at the point $z$ outside an exceptional set up to $C\log|z|$. In this article we prove that if such an approximation is made more precise, i. e. a constant $C$ decreases, then, beginning with $C=\rho/4$, the size of the exceptional set enlarges substantially. Similar results are proved for subharmonic functions of infinite order and functions subharmonic in the unit disk. These theorems improve and complement a result by Yulmukhametov.