Comparison of two norms on polynomials

Question

Let $E=R_n[X]$, $N_1(P) = \max_{0\leq k \leq n} |a_k|$ (if $P=\sum_{k=0}^n a_k X^k$) and $N_2(P) = \max_{t\in [0,1]} |P(t)|$.

What is the best $C$ we can find such that $N_1 \leq C N_2$ on $E$ ?

Answer 1

Using the explicit expression of a Vandermonde matrix, one can prove that at least, $C\leq (2\sqrt{n}^n)^{n+1}$.