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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)>=0 for all z in D. Then for all z in D, we have

0<=u(z)<=(R/(R-|z-z_0|))^2u(z_0).

See also

Harmonic Function, Harnack's Principle, Liouville's Conformality Theorem

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References

Flanigan, F. J. "Harnack's Inequality." §2.5.1 in Complex Variables: Harmonic and Analytic Functions. New York: Dover, pp. 88-90, 1983.Krantz, S. G. "The Harnack Inequality." §7.6.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 97, 1999.

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Harnack's Inequality

Cite this as:

Weisstein, Eric W. "Harnack's Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HarnacksInequality.html

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