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Markov inequalities for polynomials with restricted coefficients. (English) Zbl 1176.26007

Summary: Essentially sharp Markov-type inequalities are known for various classes of polynomials with constraints including constraints of the coefficients of the polynomials. For \(\mathbb N\) and \(\delta>0\) we introduce the class \({\mathcal F}_{n,\delta}\) as the collection of all polynomials of the form \(P(x)=\sum_{k=h}^n a_kx^k\), \(a_k\in\mathbb Z\), \(|a_k|\leq n^\delta\), \(|a_h|= \max_{h\leq k\leq n}|a_k|\). In this paper, we prove essentially sharp Markov-type inequalities for polynomials from the classes \({\mathcal F}_{n,\delta}\) on \([0,1]\). Our main result shows that the Markov factor \(2n^2\) valid for all polynomials of degree at most n on \([0,1]\) improves to \(c_\delta n\log(n+1)\) for polynomials in the classes \({\mathcal F}_{n,\delta}\) on \([0,1]\).

MSC:

26D15 Inequalities for sums, series and integrals
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