We have the following inegalities :
$c_1 n (k+1) \leq \sup_{0 \neq f \in \mathcal P_{k,n}} \frac{ \| f' \|_{[-1,1]} }{ \| f \|_{[-1,1]} } \leq c_2 n (k+1)$
Where $c_1 > 0$, $c_2 > 0$ are absolute constants and $\mathcal P_{k,n}$ is the set of all polynomials of degree at most $n$ with real coecients and with at most $k$ ($0 \leq k \leq n$) zeros in the open unit disk.