Harnack's inequality

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Harnack's inequality is an inequality arising in mathematical analysis.

Let D = D(z0,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

\sup_{x \in \omega} u(x) \le C \inf_{x \in \omega} u(x).
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