TITLE: L1 analog of Bernstein's inequality
QUESTION [7 upvotes]: Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$.  Is it true that
$$
\|q\|_1 \leq O(n) \|p\|_1
$$
where we define $\|f\|_p := \left(\int_{-1}^1 |f(x)|^pdx\right)^{1/p}$?
For reference, Bernstein's inequality shows that
$$
\|q\|_\infty \leq n\|p\|_\infty
$$
with equality at the $n$th Chebyshev polynomial, and looking at Legendre polynomials shows that
$$
\|q\|_2 \leq \sqrt{n(n+1)}\|p\|_2
$$
with equality at the $n$th Legendre polynomial.

REPLY [4 votes]: Appendix A4 of the book 

P. Borwein, T. Erdelyi, Polynomials and Polynomial inequalities, Graduate Texts in Mathematics 161, Springer

should be a good source for your question. In particular, (A.4.22) gives
$$\|P'\|_p\leq cn^2\|P\|_p,$$
for every polynomial $P$ of degree $n$ and $0<p<\infty$. Apparently, finding the best possible constant $c$ is still an open problem. 
There is also a weighted analog of the above inequality, see Theorem A.4.16., which holds for generalized polynomials, see (A.4.1) for a definition.