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If $p_n(z)$ is a polynomial with degree $n$ and $|p_n(z)|\le M$ for all $|z|<1$, then $|p_n(z)|\le M |z|^n$ for all $1\le |z| < \infty$.

It sounds reasonable. For example, take $n=3$, and $p_n(z) = 1 + \frac{z}{2} + \frac{z^2}{3}$ , then $M=1+\frac12+\frac13=1\frac56$; while $q_n(z):= \frac{p_n(z)}{z^n} = \frac13 + \frac1{2z}+\frac1{z^2}$, it does sounds get its maximum at $z=1$ and the maximum is just $M$.

But I'm a bit lost, couldn't find a way to prove it. Pls kindly suggest where can I go?

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