Abstract
After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. Kojić has shown that in the plane case both the class of quasinearly subharmonic functions and the class of regularly oscillating functions are invariant under conformal mappings. We give partial generalizations to her results by showing that in ℝn, n ≥ 2, these both classes are invariant under bi‐Lipschitz mappings.
Highlights
If D is an open set in Rn, and x ∈ D, we write δD x for the distance between the point x and the boundary ∂D of D
Let Ω be an open set in Rn, n ≥ 2
Let D be an open set in Rn, n ≥ 2
Summary
Our notation is rather standard; see, for example, 1–3 and the references therein. The Lebesgue measure in Rn, n ≥ 2, is denoted by mn. We write Bn x, r for the ball in Rn, with center x and radius r. Recall that mn Bn x, r νnrn, where νn : mn Bn 0, 1. If D is an open set in Rn, and x ∈ D, we write δD x for the distance between the point x and the boundary ∂D of D. Our constants C are nonnegative, mostly ≥ 1, and may vary from line to line
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