Let $f$ be a monic polynomial such that $\lvert f \rvert \leq 1$ if $ \lvert z \rvert \leq 1$, then $f = z^n$.

Question

Consider an holomorphic function $$f = z^n + a_{n-1}z^{n-1} + \cdots a_0 .$$ Now if $\lvert f \rvert \leq 1$ if $ \lvert z \rvert \leq 1$, then show that $f = z^n$.

My attempt, I tried substituting $z = e^{2\pi i/n}$, and checked for $n = 1, 2$, but I don't know how to proceed in general. Can anyone provide me a hint?

Answer 1

Look at the function $q(z) = z^n f(1/z)$.

Further hint if necessary:

Apply the maximum modulus principle to it.