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Let $\mathscr{H}$ denote the class of positive harmonic functions on a bounded domain $\Omega$ in ${ℝ}^N$. 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 ${\int }_{S}ud\sigma \le K{\int }_{\partial \Omega }ud\sigma$ for every $u\in \mathscr{H}\cap C\left(\overline{\Omega }\right)$. 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...