Let $p(x) = \sum_{i=0}^d c_i x^i$ be a polynomial with real coefficients of degree $d$ such that \begin{equation} |\max_{x \in [0,1]} p(x) | \leq 1 \end{equation} Can we derive an upper bound on the maximum coefficient (in absolute value), that is $$ \max_i |c_i| \leq C, $$ where $C$ is a constant ?
I can get an upper bound on $c_0$ since $ |p(0)| = |c_0| \leq 1$, but I cannot get an upper bound for the absolute value of the maximum coefficient.