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.



L1 analog of Bernstein's inequality
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.


