Point evaluation for polynomials on the circle

We study the constant Cd,p defined as the smallest constant C such that P _∞ ᵖ ≤ C P ₚᵖ holds for every polynomial P of degree d , where we consider the Lᵖ norm on the unit circle. We conjecture that Cd,p ≤ dp/2+1 for all p ≥ 2 and all degrees d . We show that the conjecture holds for all p ≥ 2 when d ≤ 4 and for all d when p ≥ 6.8.