It is a classical result that every subharmonic function, defined and ℒp-integrable for some p, 0 < p < +∞, on the unit disk 𝔻 of the complex plane ℂ is for almost all θ of the form o((1 − |𝓏|)−1/p), uniformly as 𝓏 → e𝒾θ 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 ℝ𝓃, 𝓃 ≥ 2.