Integral inequalities for algebraic and trigonometric polynomials

In this paper, we consider sharp estimates of integral functionals for functions phi defined on the semiaxis (0, a) and operators L on the set T (n) of real trigonometric polynomials f (n) of order n a parts per thousand yen 1 by the uniform norm of the polynomials. We also consider similar problems for algebraic polynomials on the unit circle of the complex plane. P. Erdos, A. Calderon, G. Klein, L. V. Taikov, and others investigated such inequalities. In particular, we show that, for 0 a parts per thousand currency sign q < a, the sharp inequality holds on the set T (n) , n a parts per thousand yen 1, for the Weyl fractional derivatives D (alpha) f (n) of order alpha a parts per thousand yen 1. For q = a (alpha a parts per thousand yen 1), this fact was proved by P.I. Lizorkin (1965). For 1 a parts per thousand currency sign q < a and positive integer alpha, the inequality was proved by L.V. Taikov (1965); however, in this case, the inequality follows from results of an earlier paper by A. P. Calderon and G. Klein (1951).