Certain inequalities in various metrics for trigonometric polynomials and their derivatives

We establish for $0<p<1$ the analog of the Bernstein–Zygmund inequality for the derivative of a trigonometric polynomial
$$ \int_{-\pi}^\pi|t_n'(x)|^pdx\leqslant c_pn^p\int_{-\pi}^\pi|t_n(x)|^pdx. $$
We prove weighted inequalities, exact in the sense of order, for trigonometric polynomials and their derivatives in various integral metrics with exponents $0<p$, $q\leqslant\infty$.