Markov-Bernstein type inequalities under Littlewood-type coefficient constraints
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Let Fn denote the set of polynomials of degree at most n with coefficients from {-1, 0, 1}. Let Gn be the collection of polynomials p of the form (equation presented) where m is an unspecified nonnegative integer not greater than n. We establish the right Markov-type inequalities for the classes Fn and Gn on [0, 1]. Namely there are absolute constants C1 > 0 and C2 > 0 such that (equation presented) and (equation presented) It is quite remarkable that the right Markov factor for Gn is much larger than the right Markov factor for Fn. We also show that there are absolute constants C1 > 0 and C2 > 0 such that (equation presented) where Ln denotes the set of polynomials of degree at most n with coefficients from {-1, 1}. For polynomials p ∈ F := U∞n=0 Fn with |p(0)| = 1 and for y ∈ [0, 1) the Bernstein-type inequality (equation presented) is also proved with absolute constants C1 > 0 and C2 > 0. This completes earlier work of the authors where the upper bound in the first inequality is obtained.