Harnack's Inequality

Let $D=D(z_0,R)$ be an Open Disk, and let $u$ be a Harmonic Function on $D$ such that $u(z)\geq 0$ for all $z\in D$. Then for all $z\in D$, we have

$$0\leq u(z)\leq \left({R\over R-\vert z-z_0\vert}\right)^2 u(z_0).$$

See also Liouville's Conformality Theorem

References

Flanigan, F. J. ``Harnack's Inequality.'' §2.5.1 in Complex Variables: Harmonic and Analytic Functions. New York: Dover, pp. 88-90, 1983.