Geodesic
On a generalization of the Bernstein–Markov inequality
Algebra i analiz, Tome 14 (2002) no. 4, pp. 36-53 
T. Erdélyi; J. Szabados. On a generalization of the Bernstein–Markov inequality. Algebra i analiz, Tome 14 (2002) no. 4, pp. 36-53. http://geodesic.mathdoc.fr/item/AA_2002_14_4_a2/
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     author = {T. Erd\'elyi and J. Szabados},
     title = {On a~generalization of the {Bernstein{\textendash}Markov} inequality},
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     pages = {36--53},
     year = {2002},
     volume = {14},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2002_14_4_a2/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that $$ \|P'Q\|_{L_p(I)}\leq c^{1+1/p}(N+M)\log(\min(N,M+1)+1)\|PQ\|_{L_p(I)} $$ for all real trigonometric polynomials $P$ and $Q$ of degree $N$ and $M$, respectively, where $0, $I:=(-\pi,\pi]$, and $c>0$ is a suitable absolute constant. Also, it is shown that $$ \|f'g\|_{L_p(J)}\leq c^{1+1/p}(N+M)^2\|fg\|_{L_p(J)} $$ for all algebraic polynomials $f$ and $g$ of degree $N$ and $M$, respectively, where $0, $J:=[-1,1]$, and $c>0$ is a suitable absolute constant. Both of the above trigonometric and algebraic results are sharp up to the factor $c^{1+1/p}$. In fact, the results are proved for the much wider classes of generalized trigonometric and algebraic polynomials.