Let $f$ be a meromorphic function on an open set $\Omega$ satisfying $Re~f \ge 0$. Prove that $f$ is in fact holomorphic.
I'm not entirely sure how to solve this, but here are some thoughts. Let $z_0$ be an isolated singularity of $f$, which, I suppose, we want to argue is in fact removable. Then $z_0$ with also be an isolated singularity of $h(z) := e^{-f(z)}$, because $z \mapsto e^{-z}$ is an entire function. But it's not hard to show that $|h(z)| \le 1$, so $h$ is bounded near $z_0$, which by Riemann's theorem means that $z_0$ is a removable singularity...But I'm not sure where to go from here...If $z \mapsto e^{-z}$ were invertible, then we'd be done, since it we could compose it with the analytic function $h$ to get back $f$, thereby proving $f$ analytic...