How to efficiently solve this question? Possibly using Max Mod Principle

Question

Let $f: {\mathbb{D}} \rightarrow {\mathbb{C}}$ be holomorphic and continuous on the closed unit disk $\bar {\mathbb{D}}$. Assume that

$|f(z)| \leq 1$ whenever $|z| = 1$ and $Im(z) > 0$; and $|f(z)| \leq 9$ whenever $|z| = 1$ and $Im(z) < 0$. Show that $|f(0)| \leq 3$.

Is it possible to solve this using the Max Mod Principle?

Answer 1

Apply the maximum modulus principle to the function $g(z) = f(z) f(-z)$ which has $|g(z)| \le 9$ everywhere on the boundary.