L-p Markov-Bernstein inequalities on arcs of the circle

Let 0 < p < z and 0 less than or equal to alpha < <beta> less than or equal to 2 pi. We prove that for trigonometric polynomials s(n) of degree less t han or equal to n, we have integral (beta)(alpha) \s'(n)(theta)\(p)[\sin(0-alpha /2)\\sin(0-beta /2)\ + (beta-alpha /n)(2)](p:2) d theta less than or equal to cn(p) integral (be ta)(alpha) \sn(theta)\(p) d theta. where c is independent of alpha, beta, n, s(n). The essential feature is th e uniformity in alpha and beta of the estimate. The result may he viewed as an L-p form of Videnskii's inequalities. (C) 2001 Academic Press.