Let $u$ be nonnegative harmonic function in $\Omega $ and $V$ an open connected subset of $\Omega $ s.t. $V\subset \subset \Omega $. Then, $$\sup_V u\leq C\inf_Vu,$$ where $C$ depending only on $V$.
remark Harnack's inequality assert that the value of a nonnegative harmonic function within $V$ are all comparable.
Question : What does it mean exactly ? (especially the remark). And in what is it specific to nonnegative harmonic functions ?