subharmonic and superharmonic functions

First let’s look at the most general definition.


Let Gn and let φ:G{-} be an upper semi-continuous function, then φ is subharmonic if for every xG and r>0 such that B(x,r)¯G (the closure of the open ball of radius r around x is still in G) and every real valued continuous functionMathworldPlanetmathPlanetmath h on B(x,r)¯ that is harmonic in B(x,r) and satisfies φ(x)h(x) for all xB(x,r) (boundary of B(x,r)) we have that φ(x)h(x) holds for all xB(x,r).

Note that by the above, the function which is identically - is subharmonic, but some authors exclude this function by definition. We can define superharmonic functions in a similar fashion to get that φ is superharmonic if and only if -φ is subharmonic.

If we restrict our domain to the complex planeMathworldPlanetmath we can get the following definition.


Let G be a region and let φ:G be a continuous function. φ is said to be subharmonic if whenever D(z,r)G (where D(z,r) is a closed disc around z of radius r) we have


and φ is said to be superharmonic if whenever D(z,r)G we have


Intuitively what this means is that a subharmonic function is at any point no greater than the averageMathworldPlanetmath of the values in a circle around that point. This implies that a non-constant subharmonic function does not achieve its maximum in a region G (it would achieve it at the boundary if it is continuous there). Similarly for a superharmonic function, but then a non-constant superharmonic function does not achieve its minumum in G. It is also easy to see that φ is subharmonic if and only if -φ is superharmonic.

Note that when equality always holds in the above equation then φ would in fact be a harmonic function. That is, when φ is both subharmonic and superharmonic, then φ is harmonic.

It is possible to relax the continuity statement above to take φ only upper semi-continuous in the subharmonic case and lower semi-continuous in the superharmonic case. The integral will then however need to be the Lebesgue integral ( rather than the Riemann integral which may not be defined for such a function. Another thing to note here is that we may take 2 instead of since we never did use complex multiplication. In that case however we must rewrite the expression z+reiθ in of the real and imaginary parts to get an expression in 2.

It is also possible generalize the range of the functions as well. A subharmonic function could have a range of {-} and a superharmonic function could have a range of {}. With this generalizationPlanetmathPlanetmath, if f is a holomorphic functionMathworldPlanetmath then φ(z):=log|f(z)| is a subharmonic function if we define the value of φ(z) at the zeros of f as -. Again it is important to note that with this generalization we again must use the Lebesgue integral.


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
  • 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title subharmonic and superharmonic functions
Canonical name SubharmonicAndSuperharmonicFunctions
Date of creation 2013-03-22 14:19:39
Last modified on 2013-03-22 14:19:39
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 12
Author jirka (4157)
Entry type Definition
Classification msc 31C05
Classification msc 31A05
Classification msc 31B05
Related topic HarmonicFunction
Defines subharmonic
Defines subharmonic function
Defines superharmonic
Defines superharmonic function