Let $d \in \mathbb{N}$. We denote by $C_{d}$ the best (smallest) constant satisfying that $$ \sup{\{ |P(z)| \colon z \in \mathbb{D} \}} \leq C_{d} \, \sup{\{ |P(x)| \colon x \in [-1,1] \}} $$ for every polynomial $P$ of degree $\leq d$ with real coefficients.
I would like to know whether these constants are known, or the best known estimations for them (with references).
Right now, I am only aware of the paper of Erdös: Some remarks on polynomials (1947).