# 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 $P_{n}$ be the class of all polynomials of degree at most $n$. If $∥⋅∥$ denotes the supremum norm on $∣z∣=1$ and $M_{p}(R)=max_{∣x∣=R}∣P(z)∣$, then for an arbitrary polynomial $P(z)=∑_{v=0}a_{v}z_{v}$ in $P_{n}$ the inequality $M_{P}(R)≤R_{n}∥P∥$ holds, with equality if and only if $a_{0}=…=a_{n–1}=0$. Given $n,k∈N$ with $0≤k≤n–1$, let $φ_{n,k}(R)$ be the largest number such that $M_{P}(R)+φ_{n,k}(R)∣a_{k}∣≤R_{n}∥P∥(R≥1)$ for all $P∈P_{n}$. Values of $φ_{n,k}(R)$ 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