# Mean values and associated measures of $\delta$-subharmonic functions

Mathematica Bohemica (2002)

• Volume: 127, Issue: 1, page 83-102
• ISSN: 0862-7959

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## Abstract

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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 $B\left(x,r\right)$, of centre $x$ and radius $r$, contained in $E$, let $ℳ\left(u,x,r\right)$ 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 of the quotient $\left(ℳ\left(u,x,s\right)-ℳ\left(u,x,t\right)\right)/\left(ℳ\left(v,x,s\right)-ℳ\left(v,x,t\right)\right)$, lie between the upper and lower limits as $r\to 0+$ of the quotient $\mu \left(B\left(x,r\right)\right)/\nu \left(B\left(x,r\right)\right)$. This enables us to use some well-known measure-theoretic results to prove new variants and generalizations of several theorems about $\delta$-subharmonic functions.

## How to cite

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Watson, Neil A.. "Mean values and associated measures of $\delta$-subharmonic functions." Mathematica Bohemica 127.1 (2002): 83-102. <http://eudml.org/doc/249014>.

@article{Watson2002,
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 $\{\mathcal \{M\}\}(u,x,r)$ 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.},
author = {Watson, Neil A.},
journal = {Mathematica Bohemica},
keywords = {superharmonic; $\delta$-subharmonic; Riesz measure; spherical mean values; superharmonic function; -subharmonic function; Riesz measure; spherical mean values},
language = {eng},
number = {1},
pages = {83-102},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Mean values and associated measures of $\delta$-subharmonic functions},
url = {http://eudml.org/doc/249014},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Watson, Neil A.
TI - Mean values and associated measures of $\delta$-subharmonic functions
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 1
SP - 83
EP - 102
AB - 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.
LA - eng
KW - superharmonic; $\delta$-subharmonic; Riesz measure; spherical mean values; superharmonic function; -subharmonic function; Riesz measure; spherical mean values
UR - http://eudml.org/doc/249014
ER -

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