# 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. $\Box$