Bernstein–Szegö inequality for Riesz derivative of trigonometric polynomials in the spaces $L_p$, $0\le p\le\infty$, with the classical value of the best constant

The Bernstein-Szegő inequality for the Weyl derivative of real order $\alpha ⩾0$ of trigonometric polynomials of degree $n$ is considered. The aim is to find values of the parameters for which the sharp constant in this inequality is equal to $n^{\alpha }$ (the classical value) in all $L_p$-spaces, $0⩽p⩽\infty$. The set of all such $\alpha$ is described for some important particular cases of the Weyl-Szegő derivative, namely, for the Riesz derivative and for the conjugate Riesz derivative, for all $n\in N$.