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On the characterization of harmonic and subharmonic functions via mean-value properties. (English) Zbl 1187.31003
The paper studies the relation between mean-value properties on the one side and harmonicity and subharmonicity on the other side. If $$G$$ is a bounded domain with sufficiently smooth boundary and $$u$$ is a continuous function on its closure $$\overline G$$, denote by $$B_u(G)$$ the mean value of $$u$$ over $$G$$, and by $$S_u(G)$$ the mean value of $$u$$ over $$\partial G$$. The first result is the following: Let $$u$$ be a continuous function on a domain $$\Omega$$. Then $$u$$ is harmonic (or subharmonic) in $$\Omega$$ if and only if $$B_u(B)=S_u(B)$$ (or $$B_u(B)\leq S_u(B)$$) for every ball $$B$$ whose closure $$\overline B$$ is contained in $$\Omega$$. The second result is the following: Let $$\Omega$$ be a bounded domain with sufficiently smooth boundary. Then there exist constants $$0<c_1 \leq 1\leq c_2 <\infty$$ such that $$c_1S_u (\Omega ) \leq B_u (\Omega ) \leq c_2 S_u(\Omega )$$ for all non-negative harmonic functions $$u\in C^1(\overline \Omega )$$. If $$c_1 =1$$ or $$c_2=1$$, then $$\Omega$$ is a ball.

##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
##### Keywords:
harmonic function; subharmonic function; mean value
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##### References:
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