Subharmonic Function

Let U subset= C be an open set and f a real-valued continuous function on U. Suppose that for each closed disk D^_(P,r) subset= U and every real-valued harmonic function h defined on a neighborhood of D^_(P,r) which satisfies f<=h on partialD(P,r), it holds that f<=h on the open disk D(P,r). Then f is said to be subharmonic on U (Krantz 1999, p. 99).

1. If f_1,f_2 are subharmonic on U, then so is f_1+f_2.

2. If f_1 is subharmonic on U and a>0 is a constant, than af_1 is subharmonic on U.

3. If f_1,f_2 are subharmonic on U, then max{f_1(z),f_2(z)} is also subharmonic on U.

See also

Barrier, Harmonic Function

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Krantz, S. G. "The Dirichlet Problem and Subharmonic Functions." §7.7 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 97-101, 1999.

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Subharmonic Function

Cite this as:

Weisstein, Eric W. "Subharmonic Function." From MathWorld--A Wolfram Web Resource.

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