Real vs complex norm for polynomials 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).
 A: 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).
