Complex Analysis and Riemann Surfaces

If $X$ is connected, then the ratio $f/g$ is constant.

If $X$ is connected, then some linear combination $f- \mu g$ (with $\mu\in \mathbb{C}$ constant) is constant.

If $X$ is compact and connected, then the ratio $f/g$ is constant.

If $X$ is compact and connected, then some linear combination $f- \mu g$ (with $\mu\in \mathbb{C}$ constant) is constant.