TITLE: $f$ is an entire function with Im $f\geq 0$
QUESTION [7 upvotes]: $f$ is an entire function with $\operatorname{Im}f \geq 0$. Then which of the followings are true:
1) $f$ is constant.
2) $\operatorname{Re}f$ is constant.
3) $f = 0$.
4) $f'$ is a non-zero constant.
That (3) & (4) are wrong can be shown by using $f(z) = i$. But I'm clueless about the remaining two options.

REPLY [6 votes]: Hints:
(1) If $f$ is entire then so is $e^{if}$
(2) $|e^{if}|=e^{-\operatorname{Im}(f)}\leq 1$
(3) Use Liouville's theorem