Lp Markov–Bernstein Inequalities on All Arcs of the Circle

Let 0<p<∞ and 0α<β2π. We prove that for n1 and trigonometric polynomials s_n of degree n, we have

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cn^p ∫^β_α |s_n(θ)|^p dθ, where c is independent of α, β, n, s_n. The essential feature is the uniformity in α,β] of the estimate and the fact that as α,β] approaches 0,2π], we recover the L_p Markov inequality. The result may be viewed as the complete L_p form of Videnskii's inequalities, improving earlier work of the second author.