Geodesic
Certain inequalities in various metrics for trigonometric polynomials and their derivatives
Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 489-498 Cet article a éte moissonné depuis la source Math-Net.Ru

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We establish for $0 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, $q\leqslant\infty$. 
@article{MZM_1975_18_4_a1,
     author = {V. I. Ivanov},
     title = {Certain inequalities in various metrics for trigonometric polynomials and their derivatives},
     journal = {Matemati\v{c}eskie zametki},
     pages = {489--498},
     year = {1975},
     volume = {18},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_18_4_a1/}
}
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V. I. Ivanov. Certain inequalities in various metrics for trigonometric polynomials and their derivatives. Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 489-498. http://geodesic.mathdoc.fr/item/MZM_1975_18_4_a1/