Asymptotic Inequalities Related to the Maximum Modulus of a Polynomial

Abstract

Let ${\mathcal{P}}_n$ be the class of all polynomials of degree at most $n$. If $\Vert \cdot \Vert$ denotes the supremum norm on $|z|=1$ and $M_p(R)=ma{x}_{|x|=R}|P(z)|$, then for an arbitrary polynomial $P(z)={\sum }_{v=0}^n{a}_v{z}^v$ in ${\mathcal{P}}_n$ the inequality $M_P(R)\le R^n\Vert P\Vert$ holds, with equality if and only if $a_0=\dots =a_{n–1}=0$. Given $n,k\in \mathbb{N}$ with $0\le k\le n–1$, let ${\phi }_{n,k}(R)$ be the largest number such that $M_P(R)+{\phi }_{n,k}(R)|a_k|\le R^n\Vert P\Vert (R\ge 1)$ for all $P\in {\mathcal{P}}_n$. Values of ${\phi }_{n,k}(R)$ for $k=0$ and $k=1$ are known since some time. We study the case $k\ge 2$.