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Mean values and associated measures of \(\delta \)-subharmonic functions. (English) Zbl 0998.31002
Summary: Let \(u\) be a \(\delta \)-subharmonic function with associated measure \(\mu \), and let \(v\) be a superharmonic function with associated measure \(\nu \), on an open set \(E\). For any closed ball \(B(x,r)\), of centre \(x\) and radius \(r\), contained in \(E\), let \(M(u,x,r)\) denote the mean value of \(u\) over the surface of the ball. We prove that the upper and lower limits as \(s,t\to 0\) with \(0<s<t\) of the quotient \[ \bigl(M(u,x,s)-M(u,x,t)\bigr)\Bigl/\bigl(M(v,x,s)-M(v,x,t)\bigr), \] lie between the upper and lower limits as \(r\to 0+\) of the quotient \(\mu (B(x,r))/\nu (B(x,r))\). This enables us to use some well-known measure-theoretic results to prove new variants and generalizations of several theorems about \(\delta \)-subharmonic functions.
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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