Point evaluation for polynomials on the circle

We study the constant ${\mathcal{C}}_{d,p}$ defined as the smallest constant $C$ such that $\mathrm{\parallel }P{\mathrm{\parallel }}_{\mathrm{\infty }}^p\le C\mathrm{\parallel }P{\mathrm{\parallel }}_p^p$ holds for every polynomial $P$ of degree $d$, where we consider the $L^p$ norm on the unit circle. We conjecture that ${\mathcal{C}}_{d,p}\le dp\mathrm{/}2+1$ for all $p\ge 2$ and all degrees $d$. We show that the conjecture holds for all $p\ge 2$ when $d\le 4$ and for all $d$ when $p\ge 6.8$.