(a) Let $\Omega \subset \mathbb{C}$ be open, $a \in \Omega$ and suppose that $D_{\rho}(a)=\{z \in \mathbb{C}:|z-a| \leqslant \rho\} \subset \Omega$. Let $f: \Omega \rightarrow \mathbb{C}$ be analytic.

State the Cauchy integral formula expressing $f(a)$ as a contour integral over $C=\partial D_{\rho}(a)$. Give, without proof, a similar expression for $f^{\prime}(a)$.

If additionally $\Omega=\mathbb{C}$ and $f$ is bounded, deduce that $f$ must be constant.

(b) If $g=u+i v: \mathbb{C} \rightarrow \mathbb{C}$ is analytic where $u, v$ are real, and if $u^{2}(z)-u(z) \geqslant v^{2}(z)$ for all $z \in \mathbb{C}$, show that $g$ is constant.