An entire function satisfying $Im(f(z)) > 0$ for all $z \in \mathbb{C}$ is constant

Question

This particular question was part of a multiple choice question asked in my quiz.

I contradicted other $3$ options but this one is true and I have no idea how to prove it.

Assume that $f$ is entire and satisfying $Im(f(z)) > 4$ for all $z \in \mathbb{C}$. Then show that $f$ is constant.

Kindly shed some light on which result I should use to solve this problem.

Answer 1

HINT: The upper half-plane is conformally equivalent to the unit disk.