This theorem is a generalization of the maximal modulus principle, but instead of requiring that the function is bounded as we approach the boundary, we only need a restriction on its growth to it to in fact be bounded in all of .
If you let (and perhaps), then you get almost exactly one version of the maximal modulus principle. In this case it turns out that need not be simply connected since that is only needed to define .
In fact the requirement that be simply connected can be eased up a bit in this theorem since it is only needed locally. So the theorem is still true if for every point there exists an open neighbourhood of such that is simply connected.
- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
|Date of creation||2013-03-22 14:12:09|
|Last modified on||2013-03-22 14:12:09|
|Last modified by||jirka (4157)|