subharmonic and superharmonic functions
First let’s look at the most general definition.
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 is superharmonic if and only if is subharmonic.
If we restrict our domain to the complex plane we can get the following definition.
Let be a region and let be a continuous function. is said to be subharmonic if whenever (where is a closed disc around of radius ) we have
and is said to be superharmonic if whenever we have
Intuitively what this means is that a subharmonic function is at any point no greater than the average 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 (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 . It is also easy to see that is subharmonic if and only if is superharmonic.
Note that when equality always holds in the above equation then would in fact be a harmonic function. That is, when is both subharmonic and superharmonic, then is harmonic.
It is possible to relax the continuity statement above to take 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 (http://planetmath.org/Integral2) rather than the Riemann integral which may not be defined for such a function. Another thing to note here is that we may take instead of since we never did use complex multiplication. In that case however we must rewrite the expression in of the real and imaginary parts to get an expression in .
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 generalization, if is a holomorphic function then is a subharmonic function if we define the value of at the zeros of 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|
|Date of creation||2013-03-22 14:19:39|
|Last modified on||2013-03-22 14:19:39|
|Last modified by||jirka (4157)|