In this paper, we consider sharp estimates of integral functionals $\int_0^{2\pi } {\phi (L|Lf_n (t)|)dt} $ for functions φ defined on the semiaxis (0, ∞) and operators L on the set T n of real trigonometric polynomials f n of order n ≥ 1 by the uniform norm $\left\| {f_n } \right\|_{C_{2\pi } } $ 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 ≤ q < ∞, the sharp inequality $\left\| {D^\alpha f_n } \right\|_{L_q } \leqslant n^\alpha \left\| {\cos t} \right\|_{L_q } \left\| {f_n } \right\|_\infty $ holds on the set T n , n ≥ 1, for the Weyl fractional derivatives D α f n of order α ≥ 1. For q = ∞ (α ≥ 1), this fact was proved by P.I. Lizorkin (1965). For 1 ≤ q < ∞ and positive integer α, 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).