Let $D=\{z \in \mathbb{C}:|z|<1\}$ and let $f: D \rightarrow \mathbb{C}$ be analytic.

(a) If there is a point $a \in D$ such that $|f(z)| \leqslant|f(a)|$ for all $z \in D$, prove that $f$ is constant.

(b) If $f(0)=0$ and $|f(z)| \leqslant 1$ for all $z \in D$, prove that $|f(z)| \leqslant|z|$ for all $z \in D$.

(c) Show that there is a constant $C$ independent of $f$ such that if $f(0)=1$ and $f(z) \notin(-\infty, 0]$ for all $z \in D$ then $|f(z)| \leqslant C$ whenever $|z| \leqslant 1 / 2 .$

[Hint: you may find it useful to consider the principal branch of the map $z \mapsto z^{1 / 2}$.]

(d) Does the conclusion in (c) hold if we replace the hypothesis $f(z) \notin(-\infty, 0]$ for $z \in D$ with the hypothesis $f(z) \neq 0$ for $z \in D$, and keep all other hypotheses? Justify your answer.