Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1. (English) Zbl 1345.26021
Summary: For \(n \in {\mathbb N}\), \(L > 0\), and \(p \geq 1\) let \(\kappa_p(n,L)\) be the largest possible value of \(k\) for which there is a polynomial \(P \not \equiv 0\) of the form
\[
P(x) = \sum_{j=0}^n{a_jx^j}, |a_0| \geq L \Bigl( \sum_{j=1}^n{|a_j|^p} \Bigr)^{1/p}, \; a_j \in {\mathbb C},
\]
such that \((x-1)^k\) divides \(P(x)\). For \(n \in {\mathbb N}\), \(L > 0\), and \(q \geq 1\) let \(\mu_q(n,L)\) be the smallest value of \(k\) for which there is a polynomial \(Q\) of degree \(k\) with complex coefficients such that
\[
|Q(0)| > \frac 1L \Bigl( \sum_{j=1}^n{|Q(j)|^q} \Bigr)^{1/q}.
\]
We find the size of \(\kappa_p(n,L)\) and \(\mu_q(n,L)\) for all \(n \in {\mathbb N}\), \(L > 0\), and \({1 \leq p,q \leq \infty}\). The result about \(\mu_\infty(n,L)\) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even in that special case.
MSC:
26C10 | Real polynomials: location of zeros |
41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
11C08 | Polynomials in number theory |
Keywords:
polynomials; restricted coefficients; order of vanishing at 1; Coppersmith-Rivlin type inequalitiesReferences:
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