An extremal problem for polynomials with a prescribed zero

Abstract:

Let ${\mathcal {P}_{n,b}}$ denote the class of all polynomials ${p_n}(z)$ of degree at most $n$ in $z$ which satisfy ${\max _{|z| = 1}}|{p_n}(z)| = 1$, and $|{p_n}(1)| = b,0 \leqq b < 1$. Let $c \in (0,n]$, and set \[ {\mu _b}(c,n) = \sup \limits _{{p_n} \in {\mathcal {P}_{n,b}}} \{ \min \limits _{|z| = 1 - c/n} |{p_n}(z)|\} .\] Upper estimates for ${\mu _b}(c,n)$ are obtained.