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MATHEMATICA BOHEMICA, Vol. 127, No. 1, pp. 83-102 (2002)
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# Mean values and associated measures of $\delta$-subharmonic functions

## Neil A. Watson

* Neil A. Watson*, Department of Mathematics, University of Canterbury, Christchurch, New Zealand, e-mail: ` N.Watson@math.canterbury.ac.nz`

**Abstract:**
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 $\MM (u,x,r)$ 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 $(\MM (u,x,s)-\MM (u,x,t))/(\MM (v,x,s)-\MM (v,x,t))$, 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.

**Keywords:** superharmonic, $\delta$-subharmonic function, Riesz measure, spherical mean values

**Classification (MSC2000):** 31B05

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