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I am trying to solve the following problem:

Let $P(z)$ be a polynomial in $z $of degree $n$. Show that if $|P(z)| ≤ M$ for $|z| = 1$ then $|P(z)| ≤ M|z|^n$ for all $|z| ≥ 1$.

But I am having some trouble. I know that if $P(z)=a_nz^n+\dots+a_0$, then

$$a_k=\frac{k!}{2\pi i}\int_{|z|=1}\frac{p(\zeta)}{\zeta^{k+1}}d\zeta$$

and therefore $$|a_k|\le k!M$$

but after this I am stuck. Any help would be apreciated.

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1 Answer 1

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Hint: $Q(z)=z^{n} P(\frac 1 z)$ is a polynomial. Apply MMP to this on the unit disk.

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