Abstract
Using the notion of inferior mean due to M. Heins, we establish two inequalities for such a mean relative to a positive harmonic function defined on the open unit ball or half-space in .
Highlights
Wu showed in [4] that for a positive harmonic function in the unit disc, one has in most cases inequality, while equality occurs for functions whose boundary measures are absolutely continuous
The bound is achieved for functions whose boundary measures, for example, are purely singular
For any positive harmonic function u on Ω with boundary measure μ, there exists the following inequality: IM(u) ≤ μ. Equality occurs for those u whose boundary measures μ are absolutely continuous, when the inferior mean is attained along boundaries of Aδ not equal to S as δ → 0
Summary
Wu showed in [4] that for a positive harmonic function in the unit disc, one has in most cases inequality, while equality occurs for functions whose boundary measures are absolutely continuous. She showed that there exists a nonzero lower bound of the lim inf for this class of functions in the disc. For any positive harmonic function u on Ω with boundary measure μ, there exists the following inequality: IM(u) ≤ μ Equality occurs for those u whose boundary measures μ are absolutely continuous, when the inferior mean is attained along boundaries of Aδ not equal to S as δ → 0. The proofs rely on Sard’s theorem (see [3]), and inequality (2.5) obtained below
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