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This question might be obvious, but it's been stumping me for the moment. Let $p(z)=\sum^n_{k=0} a_kz^k$ be a complex polynomial. Is it true that $$\max_{k\in\{0,\ldots,n\}}|a_k| \leq \max_{|z|=1}|p(z)|?$$ I'm guessing that it's true, but I can't find a satisfactory proof. I would appreciate it if someone could explain why it's true, or if it is not, provide a counterexample.

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Note that $a_k = {p^{(k)}(0) \over k!}$. Then Cauchy's estimate gives $|p^{(k)}(0)| \le {k! M_R \over R^k}$, where $M_R = \max_{|z| = R} |p(z)|$. In this case, we have $R = 1$, and we get the desired result.

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