This is an exercise from Conway. I managed to solve (a), but am stuck at (b). How can I find a formula for the function f with just the value at z=0? I have suspicions that f must be some finite blaschke product, but can't prove it... Could anyone please help me?
(a) Apply the maximum principle to $$g(z) = f(z) \prod_{k=1}^n \frac{1-\overline{z_k}z}{z-z_k}$$ observing that as $z$ approaches the boundary of $D$, $$|g(z)| \le M \prod_{k=1}^n \left|\frac{1-\overline{z_k}z}{z-z_k}\right| \to M$$ Hence $|g|\le M$ in $D$.
(b) If $f(0)=Me^{i\alpha}z_1\cdots z_n$, then $g(0)=(-1)^nMe^{i\alpha}$. The maximum principle says that only constant functions can attain the maximum of their modulus inside of the domain. Hence $g\equiv (-1)^nMe^{i\alpha}$, which yields a formula for $f$.