[Math] Real part of holomorphic function cannot have a maximum

Question

I am trying to prove that if $f$ is a holomorphic function from a domain $U$ to $\mathbb{C}$, and the real part has an interior local maximum at a point $a$ in $U$, then $f$ is a constant. I am new to complex analysis, but I was thinking maybe I need to use some variant of the local maximum principle?

Thanks.

Answer 1

The real (as well as imaginary) part of a holomorphic function is harmonic, and there is an analogous maximum principle for harmonic functions.