Self-improving estimates of growth of subharmonic and analytic functions

TL;DR

This paper investigates how estimates for subharmonic and analytic functions can be improved or extended, providing explicit relations between bounds involving distances to certain sets, and compares different methodological approaches.

Contribution

It introduces explicit formulas relating bounds of subharmonic functions to those of analytic functions, building on and comparing Domar's classical work with new results.

Findings

Derived explicit expressions for bounds of functions in terms of distance functions.

Extended Domar's classical results to new settings for subharmonic and analytic functions.

Compared different approaches for estimating growth of these functions.

Abstract

Given a bounded open subset $\Omega$ and closed subsets $A,B$ of $\mathbb{R}^k$, we discuss when an estimate $u(x)\le g(dist(x,A\cup B))$, $x\in\Omega\setminus(A\cup B)$, for a function $u$ subharmonic on $\Omega\setminus B$, implies that $u(x)\le h(dist(x,B))$, $x\in\Omega\setminus B$, where $g,h:(0,\infty)\to (0,\infty)$ are decreasing functions and $g(0^+)=h(0^+)=\infty$. We seek for explicit expressions of $h$ in terms of $g$. We give some results of this type and show that Domar's work (On the existence of a largest subharmonic minorant of a given function, Ark. Mat., 3 (1957), pp. 429-440) permits one to deduce other results in this direction. Then we compare these two approaches. Similar results are deduced for estimates of analytic functions.