  We begin by recalling the definition of nonnegative quasinearly subharmonic
functions on locally uniformly homogeneous spaces. Recall that these spaces and
this function class are rather general: among others subharmonic,
quasisubharmonic and nearly subharmonic functions on domains of Euclidean
spaces ${\mathbb{R}}^n$, $n\geq 2$, are included. The following result of
Gehring and Hallenbeck is classical: 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 both Pavlovi\'c and Riihentaus have given related and partly
more general results on domains of ${\mathbb{R}}^n$, $n\geq 2$. Now we extend
one of these results to quasinearly subharmonic functions on locally uniformly
homogeneous spaces.
