On an inequality related to the radial growth of subharmonic functions

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 $θ$ of the form $o((1-| z|)^{-1/p})$, uniformly as $z\to e^{iθ}$ in any Stolz domain. Recently Pavlović gave a related integral inequality for absolute values of harmonic functions, also defined on the unit disk in the complex plane. We generalize Pavlović's result to so called quasi-nearly subharmonic functions defined on rather general domains in ${\mathbb {R}}^n$, $n\geq 2$.