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Mean value type inequalities for quasinearly subharmonic functions. (English) Zbl 1271.31008

Let \(A\subset (0,\infty )\), let \(B(x,r)\) denote the open ball in \(\mathbb{R}^{n}\) of centre \(x\) and radius \(r\), and let \(\widehat{\lambda }\) denote the normalised volume measure on \(B(x,r)\). If \(0\in \overline{A}\) and \(u:\Omega \rightarrow [ -\infty ,\infty )\) is an upper semicontinuous function on an open set \(\Omega \) in \(\mathbb{R}^{n}\) such that \(u(x)\leq \int_{B(x,r)}ud\widehat{\lambda }\) whenever \(\overline{B}(x,r)\subset \Omega \) and \(r\in A\), then \(u\) is known to be subharmonic on \(\Omega \). A sample result in this paper asks for conditions on \(A\) so that, if \(u:\Omega \rightarrow [ 0,\infty )\) is locally integrable and satisfies \(u(x)\leq K\int_{B(x,r)}ud\widehat{\lambda }\) whenever \(\overline{B} (x,r)\subset \Omega \) and \(r\in A\), for some \(K\geq 1\), then \(u\) necessarily satisfies an inequality of this form for all \(\overline{B}(x,r)\subset \Omega \). It is shown that the right condition is that there exists \(C>1\) such that \([t/C,t]\cap A\neq \emptyset \) for every \(t>0\).

MSC:

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
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