Markov Inequalities for Polynomials with Restricted Coefficients

Abstract

Essentially sharp Markov-type inequalities are known for various classes of polynomials with constraints including constraints of the coefficients of the polynomials. For ℕ and δ>0 we introduce the class ℱn,δ as the collection of all polynomials of the form P(x)=∑k=hnakxk, ak∈ℤ, |ak|≤nδ, |ah|=max⁡h≤k≤n|ak|. In this paper, we prove essentially sharp Markov-type inequalities for polynomials from the classes ℱn,δ on [0,1]. Our main result shows that the Markov factor 2n2 valid for all polynomials of degree at most n on [0,1] improves to cδnlog⁡(n+1) for polynomials in the classes ℱn,δ on [0,1].