Bernstein–Szegö inequality for fractional derivatives of trigonometric polynomials

On the set $\mathscr F_n$ of trigonometric polynomial of degree $n\ge1$ with complex coefficients, we consider the Szegö operator $D^\alpha_\theta$ defined by the relation $D^\alpha_\theta f_n(t)=\cos\theta D^\alpha f_n(t)-\sin\theta D^\alpha\widetilde f_n(t)$ for $\alpha,\theta\in\mathbb R$, $\alpha\ge0$; where $D^\alpha f_n$ and $D^\alpha\widetilde f_n$ are the Weyl fractional derivatives of (real) order $\alpha$ of the polynomial $f_n$ and its conjugate polynomial $\widetilde f_n$. In particular, we prove that, if $\alpha\ge n\ln2n$, then, for any $\theta\in\mathbb R$, the sharp inequality $\|\cos\theta D^\alpha f_n-\sin\theta D^\alpha\widetilde f_n\|_{L_p}\le n^\alpha\|f_n\|_{L_p}$ holds in the spaces $L_p$ for all $p\ge0$ on the set $\mathscr F_n$. For classical derivatives (of integer order $\alpha\ge1$), this inequality was obtained by Szegö (1928) in the uniform norm $(p=\infty)$ and by Zygmund (1931–1935) for $1\le p<\infty$. A. I. Kozko (1998) proved this inequality for fractional derivatives of (real) order $\alpha\ge1$ and $1\le p\le\infty$.