Harnack's inequality

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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 $z \in D$ :

$0\le f(z)\le \left( \frac{R}{R-\left|z-z_0\right|}\right)^2f(z_0).$

For general domains in $\mathbf{R}^n$ the inequality can be stated as follows: If $u (x)$ is twice differentiable, harmonic and nonnegative, $ω$ is a bounded domain with $\bar{\omega} \subset \Omega$ , then there is a constant $C$ which is independent of $u$ such that