This problem is totally out of my ability. Not even sure what it is talking about. Somebody please help me to solve this...
A finite Blaschke product of degree $n-1$ is a function of the form
$ \ B_{n-1} (z) = e^{i \varphi} \prod_{k=1}^{n-1} \frac{z - \alpha_k}{1 - \bar{\alpha}_k z} ,~~ | \alpha_k | < 1 \ $
(a) Explain why $B_{n-1}$ is analytic inside and on the unit circle $C = \{ z : | z | = 1 \}$.
(b) Show that $| B_{n-1} (z) | = 1$ at all points $z$ on the unit circle. [Hint: Show that each factor in the product has absolute value $1$ if $z = e^{i \theta}$.]
(c) Suppose $g(z)$ is a function that is analytic inside and on the unit circle $C$ and matches $B_{n-1}$ at $n$ points inside $C$:
$ \ g( \lambda_j ) = B_{n-1} ( \lambda_j ) ,~~j = 1, \ldots , n ,~~~| \lambda_j | < 1. \ $
Use Rouche's theorem to show that $| g(z) | \geq | B_{n-1} (z) |$ at some point $z$ on the unit circle.