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Markov- and Bernstein-Type Inequalities for Polynomials with Restricted Coefficients

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Abstract

The Markov-type inequality\(||p\prime ||_{[0,1]} \leqslant cn\log (n + 1)||p||_{[0,1]} \) is proved for all polynomials of degree at most n with coefficients from {-1,0,1} with an absolute constant c. Here ‖·‖[0,1] denotes the supremum norm on [0,1]. The Bernstein-type inequality\(|p\prime (y)| \leqslant \frac{c} {{(1 - y)^2 }}||p||_{[0,1]} ,y \in [0,1) \) is shown for every polynomial p of the form \(p(x) = \sum\limits_{j = m}^n {a_j x^j } ,|a_m | = 1,|a_j | \leqslant 1,a_{j \in } \mathbb{C} \) The inequality \(|p\prime (y)| \leqslant \frac{c} {{(1 - y)}}\log \left( {\frac{2} {{1 - y}}} \right)||p||_{[0,1]} ,y \in [0,1) \) is also proved for every analytic function p on the open unit disk D that satisfies the growth condition \(|p(0)| = 1,|p(z) \leqslant \frac{1} {{1 - |z|}},z \in D \)

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Authors
  1. Peter Borwein
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  2. Tamás Erdélyi
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Borwein, P., Erdélyi, T. Markov- and Bernstein-Type Inequalities for Polynomials with Restricted Coefficients. The Ramanujan Journal 1, 309–322 (1997). https://doi.org/10.1023/A:1009761214134

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