On an inequality related to the radial growth of subharmonic functions

It is a classical result that every subharmonic function, defined and ${\mathcal{L}}^p$-integrable for some $p$, $0<p<+\mathrm{\infty }$, on the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ is for almost all $\theta$ of the form $o((1-\mathrm{\mid }z\mathrm{\mid }{)}^{-1\mathrm{/}p})$, 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 ${\mathbb{R}}^n$, $n\ge 2$.