Entire function and boundedness

Question

I've read a proposition that

If $f$ is entire and Im$(f(z))\gt0$ for $z \in \mathbb C$, then $f$ has to be a constant.

I am wondering why it is true. And what does Im$(f(z))\gt0$ imply? Does that mean $f(z)$ is a constant and thus $f$ is a constant?

Answer 1

Hint: Consider the entire function $g(z)=e^{i f(z)}$ and use the Liouville's Theorem

Answer 2

The imagenary part of $f$ is harmonic. If it is also positive, you can apply Harnack's inequality to it on larger and larger balls, and then take the radius to infinity to see that it is constant.

Answer 3

Of course not: Consider $i(1+z^2).$