Prooving that a complex function is constant

Question

Let $f(z)=u(z)+v(z)i$, $u(z) \leq 0 , \forall z \in \mathbb{C}$, and $f(z)$ is entire. Then $f(z)$ is constant. The hint is "use the Liouville theorem". I tried , but i need prove which f is limited first, and i don't know how.

Answer 1

Let $g$ be a map (a Moebius transformation) that sends the open half-plane $\Re(z)<0$ to the open unit disk biholomorphically. Then $g\circ f$ is a bounded holomorphic function and so is constant. Since $g$ is a bijection, $f$ must be constant.