In "Interpolations by bounded analytic functions and the corona problem", Carleson introduced Carleson measures (for Hardy spaces) and proved the famous embedding theorem according to which $\mu$ is a Carleson measure iff the inclusion $$H^p(\mathbb{D})\hookrightarrow L^p(\mathbb{D},\mu)$$is continuous. In particular, he proved $$\int_{\mathbb{D}}|f(\zeta)|^pd\mu(\zeta)\le C\cdot B(\mu)\int_{\mathbb{T}}|f(e^{i\vartheta})|^p\frac{d\vartheta}{2\pi} $$
Where $C$ is an absolute constant and $B(\mu)$ is the Carleson constant of $\mu$ (i.e. the smallest constant $B$ for which for every we have $\mu\left(\{re^{i\vartheta}:1-h\le r<1; |\vartheta-\vartheta_0|<h\}\right)\le B h$ for every choice of $h,\vartheta_0$)
Has any work been done on the best possible value of $C$? An easy argument$^1$ implies $C\ge \pi$ but I have not been able to find a decent upper bound.
$^1$:just take $\mu:\mu(A)=|A\cap [-1,1]|$ and see that Carleson's embedding theorem reduces to the Fejer-Riesz inequality with constant $C/2\pi$
