Maximum modulus principle

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 {f\in{\mathcal H}(U)}. Then if {z_0\in U} has {|f(z_0)|\ge|f(z)|} for every {z\in U}, then {f} is constant.

Proof: If {|f|} has a local maximum at {z_0}, then in a small ball around {z_0}, consider the image of {|f|}. It looks something like {(f(z_0)-\varepsilon, f(z_0)]}. 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 {B_r(z_0)} under {f} is completely contained in {\overline{B_{f(z_0)}(0)}}, 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 {f} were in fact constant. \Box


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