$f$ 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.