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?