Markov-bernstein Type Inequality for Trigonometric Polynomials with Respect to Doubling Weights on [−ω, Ω]

Various important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc. inequalities, have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condition. Here, based on a recently proved Bernstein-type inequality by D.S. Lubinsky, we establish Markov-Bernstein type inequalities for trigonometric polynomials with respect to doubling weights on [−ω, ω]. Namely, we show the theorem below. Theorem. Let p ∈ [1,∞) and ω ∈ (0, 1/2]. Suppose W (arcsin((sinω) cos t)) is a doubling weight. Then there is a constant C depending only on p and the doubling constant L so that