Maximum modulus principle
August 23, 2012 Leave a comment
Here’s a fact you probably never noticed: Holomoprhic functions have no local maxima. Okay, constant functions do, but those are lame.
Theorem 1 (Maximum modulus principle) Let . Then if has for every , then is constant.
Proof: If has a local maximum at , then in a small ball around , consider the image of . It looks something like . We don’t really care about the lower bound. The important point is that the upper bound is attained. This tells us that, the image of the small ball under is completely contained in , and that it touches the boundary. Thus, the image of this ball has a boundary and thus cannot be an open set. But by the open mapping theorem, this is only possible if were in fact constant.