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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.
##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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