Polynomial of same degree 
Let $p(z)$ and $q(z)$ are two polynomial of same degree and zeroes of $p(z)$ and $q(z)$ are inside open unit disc, $|p(z)|=|q(z)|$ on unit circle then show that $p(z)=\lambda q(z)$ where $|\lambda|=1$.

Please just give a hint not the whole solution. Thank you.

Let $d$ the common degree of $P$ and $Q$. Define $P_1(z):=z^dP(1/z)$ and $P_2(z):=z^dQ(1/z)$. These polynomial doesn't vanish in the unit disk. We apply maximum modulus principle to $P_1/P_2$ and $P_2/P_1$. This gives that $P_1/P_2$ is constant (otherwise we get a contradiction because $P_1/P_2$ would be a holomorphic function with a constant modulus on a connected open set, hence would be itself constant).
Note that the fact that $P$ and $Q$ have the same degree is necessary. Indeed, $P(z)=z$ and $Q(z)=z^2$ have their root in the open unit disk, the same modulus on the unit sphere but of course are not equal up to a constant. 
