On a generalization of the Bernstein–Markov inequality

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 $0<p\leq\infty$, $I:=(-\pi,\pi]$, 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 $0<p\leq\infty$, $J:=[-1,1]$, 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.