Bernstein-Szego inequality for trigonometric polynomials in the space $L_0$

Inequalities of the form $\|f_n^{(\alpha)}\cos\theta+\tilde{f}_n^{(\alpha)}\sin\theta\|_p\le B_n(\alpha,\theta)_p \|f_n\|_p$ for classical derivatives of order $\alpha\in\mathbb{N}$ and Weyl derivatives of real order $\alpha\ge 0$ of trigonometric polynomials $f_n$ of order $n\ge 1$ and their conjugates for real $\theta$ and $0\le p\le \infty$ are called Bernstein–Szegő inequalities. They are generalizations of the classical Bernstein inequality ($\alpha=1$, $\theta=0$, $p=\infty$). Such inequalities have been studied for more than 90 years. The problem of studying the Bernstein–Szegő inequality consists in analyzing the properties of the best (the smallest) constant $B_n(\alpha,\theta)_p$, its exact value, and extremal polynomials for which this inequality turns into an equality. G. Szegő (1928), A. Zygmund (1933), and A. I. Kozko (1998) showed that, in the case $p\ge 1$ for real $\alpha\ge 1$ and any real $\theta$, the best constant $B_n(\alpha,\theta)_p$ is $n^\alpha$. For $p=0$, Bernstein–Szegő inequalities are of interest at least because the constant $B_n(\alpha,\theta)_p$ is the largest for $p=0$ over $0\le p\le\infty$. In 1981, V. V. Arestov proved that, for $r\in\mathbb{N}$ and $\theta=0$, the Bernstein inequality is true with the constant $n^r$ in the spaces $L_p$, $0\le p<1$; i.e., $B_n(r,0)_p=n^r$. In 1994, he proved that, for $p=0$ and the derivative of the conjugate polynomial of order $r\in\mathbb{N}\cup\{0 \}$, i.e., for $\theta=\pi/2$, the exact constant grows exponentially in $n$; more precisely, $B_n(r,\pi/2)_0=4^{n+o(n)}$. In two recent papers of the author (2018), a similar result was obtained for Weyl derivatives of positive noninteger order for any real $\theta$. In the present paper, we prove that the formula $B_n(\alpha,\theta)_0=4^{n+o(n)}$ holds for derivatives of nonnegative integer orders $\alpha$ and any real $\theta\neq \pi k,\,k\in\mathbb{Z}$.