On an inequality related to the radial growth of subharmonic functions

TL;DR

This paper generalizes a classical inequality for subharmonic functions on the unit disk to a broader class called quasi-nearly subharmonic functions in higher dimensions, providing new insights into their boundary behavior.

Contribution

It extends Pavlović's integral inequality from harmonic functions to quasi-nearly subharmonic functions on general domains in higher-dimensional spaces.

Findings

Established a generalized integral inequality for quasi-nearly subharmonic functions.

Extended boundary growth estimates to higher dimensions.

Provided a framework for analyzing subharmonic function behavior in complex and real domains.

Abstract

It is a classical result that every subharmonic function, defined and ${\mathcal{L}}^p$-integrable for some $p$, $0<p<+\infty$, on the unit disk $\mathbb{D}$ of the complex plane ${\mathbb{C}}$ is for almost all $\theta$ of the form $o((1-| z|)^{-1/p})$, uniformly as $z\to e^{i\theta}$ in any Stolz domain. Recently Pavlovi\'c gave a related integral inequality for absolute values of harmonic functions, also defined on the unit disk in the complex plane. We generalize Pavlovi\'c's result to so called quasi-nearly subharmonic functions defined on rather general domains in ${\mathbb {R}}^n$, $n\geq 2$.