Suppose that $p(z) = a_nz^n+\cdots+a_0$ and it has maximum modulus $1$ on the boundary of the unit disk. Suppose that $p(z) = a_nz^n+\cdots+a_0$ and it has maximum modulus $1$ on the boundary of the unit disk, show that $|p(z)| \leqslant max\{1,|z|^{n}\}$. How to show that $|p(z)| \leqslant |z|^n$?

Note that $q(z) = z^n p(z^{-1})$ is a polynomial. What do you know about $\lvert q(z) \rvert$ when $\lvert z\rvert=1$? Using the maximum modulus principle, what does that say about $\lvert p(z^{-1}) \rvert$ if $0 < \lvert z \rvert\leq 1$?
