# Complex polynomial and the unit circle

Given a polynomial $P(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0$, such that
$\max_{|z|=1} |P(z)| = 1$

Prove: $P(z) = z^n$

Hint: Use cauchy derivative estimation $$|f^{(n)} (z_0)| \leq \frac{n!}{r^n} \max_{|z-z_0|\leq r} |f(z)|$$ and look at the function $\frac{P(z)}{z^n}$

It seems to be maximum principle related, but I can't see how to use it and I can't understand how to use the hint.

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Let $g(z):=z^nP\left(\frac 1z\right)$. It's a polynomial whose leading term is $a_0$ and constant coefficient is $1$. We have that $g(0)=1$ and $\max_{|z|=1}|g(z)|=1$, hence by maximum modulus principle, $g$ is constant equal to $1$. This gives the wanted result.