State Liouville's Theorem. Prove it by considering

$$\int_{|z|=R} \frac{f(z) d z}{(z-a)(z-b)}$$

and letting $R \rightarrow \infty$.

Prove that, if $g(z)$ is a function analytic on all of $\mathbb{C}$ with real and imaginary parts $u(z)$ and $v(z)$, then either of the conditions:

$$\text { (i) } u+v \geqslant 0 \text { for all } z \text {; or (ii) } u v \geqslant 0 \text { for all } z \text {, }$$

implies that $g(z)$ is constant.