complex analysis - entire function with ist imaginary part positive

$f$ is an entire function, and it satisfies $f(\mathbb{C}) \subseteq \{z \in \mathbb{C} \mid \operatorname{Im} z > 0\}$. Show that $f$ is constant.

I want to take advantage of the Liouville's Theorem, but I just can't figure out the relationship between its image part with its module.

Answer

Consider

$$g(z)=\frac{1}{f(z)+i}.$$ Clearly it is entire, and $$\left|g(z)\right|=\frac{1}{|f(z)+i|}\leq\frac{1}{1}=1.$$ Thus, $g(z)$ is constant. Applying Liouville's theorem, $g(z)$ is constant, implying $f(z)$ is constant.

NOTE: This argument can be used to show that $f(\Bbb C)$ is dense in $\Bbb C$ by using a proof by contradiction.