HARNACK'S INEQUALITY

'Harnack's inequality' is an inequality arising in mathematical analysis.
Let $D=D(z\_0,R)$ be an open disk and let ''f'' be a harmonic function on ''D'' such that ''f(z)'' is non-negative for all $z\; in\; D$. Then the following inequality holds for all $z\; in\; D$:
:
ight|}
ight)^2f(z_0).
For general domains in $mathbf\{R\}^n$ the inequality can be stated as follows: If $u(x)$ is twice differentiable, harmonic and
nonnegative, $omega$ is a bounded domain with , then there is a constant $C$ which is independent of $u$ such that
:$sup\_\{x\; in\; omega\}\; u(x)\; le\; C\; inf\_\{x\; in\; omega\}\; u(x)$.