Let $f:U\rightarrow \mathbb{C}$ holomorphic no constant, then $\mbox{Re}f+\mbox{Im}f$ has no minimum in $U$.

Question

Let $U\in\mathbb{C}$ be an open and connected set and let $f:U\rightarrow \mathbb{C}$ holomorphic. Suppose that $f$ is not constant. Show that $\mbox{Re}f+\mbox{Im}f$ has no minimum in $U$.

Remark: I think I should use the Maximum Modulus Principle but I don't see how it can be used.

Answer 1

Hint: Prove by contradiction. First prove $\mbox{Re}f+\mbox{Im}f$ is harmonic by showing it satisfies Laplacian equation. Then prove bounded harmonic function $u$ must be constant by the fact that $e^{-u-iv}$ ($v$ is harmonic conjugate of $u$) is bounded entire.