p

Abstract. Various important weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc. inequalities, have been proved re-cently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most of the cases this minimal assumption is the doubling condi-tion. 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