# Asymptotic Inequalities Related to the Maximum Modulus of a Polynomial

### C. Frappier

École Polytechnique de Montréal, Canada### M.A. Qazi

École Polytechnique de Montréal, Canada

## Abstract

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

## Cite this article

C. Frappier, M.A. Qazi, Asymptotic Inequalities Related to the Maximum Modulus of a Polynomial. Z. Anal. Anwend. 15 (1996), no. 3, pp. 474–758

DOI 10.4171/ZAA/726