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. (Notation: $\mathbb{D}$ is the open unit disk of $\mathbb{C}$.)


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), which I found in the paper of Aron, Beauzamy and Enflo: "Polynomials in Many Variables: Real vs Complex Norms" (1993).



Real vs complex norm for polynomials
Let $K=[-1,1]$ and let $\Omega$ be its complement in $\mathbb C$. By the Bernstein's lemma (or Bernstein-Walsh lemma), see Theorem 5.5.7 in T. Ransford, Potential theory in the complex plane, Cambridge, 1995,
$$|P\_{d}(z)|\leq e^{dg\_{\Omega}(z)}\|P\_{d}\|\_{K},\qquad z\in \Omega,$$
where $g\_{\Omega}(z)$ denotes the Green function of $\Omega$ with a logarithmic singularity at infinity, whose explicit expression is
$$g\_{\Omega}(z)=\log|z+\sqrt{z^{2}-1}|.$$
It is easy to check that
$$\max\_{|z|=1} g\_{\Omega}(z)=g\_{\Omega}(i)=\log(1+\sqrt{2})$$
thus
$$\|P\_{d}\|\_{\mathbb D}\leq (1+\sqrt{2})^{d}\|P\_{d}\|\_{K}.$$
Concerning sharpness, the theorem referenced above also states that there exists a polynomial $Q\_{d}$ (a so-called Fekete polynomial) of degree $d$ such that
$$ a\_{d}^{cd}(1+\sqrt{2})^{d}\|Q\_{d}\|\_{K}\leq\|Q\_{d}\|\_{\mathbb D},$$
where $a\_{d}<1$ tends to 1 as $d\to\infty$ and $c$ is some constant
that could be explicitly computed ($a\_{d}$ involves the $d$-th diameter of $K$ and the constant $c$ is given in terms of the Harnack distance for $\Omega$, see the theorem for details).


