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Suppose $p(z)$ is a polynomial of degree $n$ having no zeros in $|z|<1.$ Then for any $R\geq1$ we have a result $ \max_{|z|=R}|p(z)|\leq \frac{1+R^n}{2}max_{|z|=1}|p(z)|,\;\;R\geq 1.$

I could not find any literature on the similar $ (\leq)$ inequality when $R<1. $ Can I expect some input?

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    $\begingroup$ Trivially for $R<1$, $\max_{|z|=R}|p(z)|\leq\max_{|z|=1}|p(z)|$. This is unimprovable (think of contant) $\endgroup$ May 9, 2020 at 15:05
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    $\begingroup$ Are you trying to reinvent Schwarz lemma? If $a_0=\dots=a_{k-1}=0$, then $\max_{|z|=R}|p(z)|\leq R^k\max_{|z|=1}|p(z)|$ $\endgroup$ May 9, 2020 at 16:41

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