(a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $|f(z)| \leqslant 1+|z|^{1 / 2}$ for every $z$. Prove that $f$ is constant.

(b) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an analytic function such that $\operatorname{Re}(f(z)) \geqslant 0$ for every $z$. Prove that $f$ is constant.