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Keywords:
superharmonic; $\delta $-subharmonic; Riesz measure; spherical mean values
superharmonic; $\delta $-subharmonic; Riesz measure; spherical mean values
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 ${\mathcal M}(u,x,r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s,t\rightarrow 0$ with $0<s<t$ of the quotient $({\mathcal M}(u,x,s)-{\mathcal M}(u,x,t))/({\mathcal M}(v,x,s)-{\mathcal M}(v,x,t))$, lie between the upper and lower limits as $r\rightarrow 0+$ 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.
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