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On an inequality related to the radial growth of quasinearly subharmonic functions in locally uniformly homogeneous spaces

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 $θ$ of the form $o((1-| z|)^{-1/p})$, uniformly as $z\to e^{iθ}$ in any Stolz domain. Recently both Pavlović 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.