Let ${\Cal H}$ denote the class of positive harmonic functions on a bounded domain $\Omega$ in $\Bbb R^N$. Let $S$ be a sphere contained in $\overline{\Omega}$, and let $\sigma$ denote the $(N-1)$-dimensional measure. We give a condition on $\Omega$ which guarantees that there exists a constant $K$, depending only on $\Omega$ and $S$, such that $\int_Su\,d\sigma \le K\int_{\partial\Omega}u\,d\sigma$ for every $u\in {\Cal H}\cap C(\overline{\Omega})$. If this inequality holds for every such $u$, then it also holds for a large class of non-negative subharmonic functions. For certain types of domains explicit values for $K$ are given. In particular the classical value $K=2$ for convex domains is slightly improved.
@article{118990,
author = {M. P. Aldred and David H. Armitage},
title = {Inequalities for surface integrals of non-negative subharmonic functions},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {39},
year = {1998},
pages = {101-113},
zbl = {0938.31001},
mrnumber = {1622990},
language = {en},
url = {http://dml.mathdoc.fr/item/118990}
}
Aldred, M. P.; Armitage, David H. Inequalities for surface integrals of non-negative subharmonic functions. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) pp. 101-113. http://gdmltest.u-ga.fr/item/118990/
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