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 $z\in D$:

$0\leq f(z)\leq \left({\frac {R}{R-\left|z-z_{0}\right|}}\right)^{2}f(z_{0}).$

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 ${\bar {\omega }}\subset \Omega$, then there is a constant $C$ which is independent of $\Omega$ such that $\sup _{x\in \omega }u(x)\leq C\inf _{x\in \omega }u(x)$.