Point evaluation for polynomials on the circle

Abstract We study the constant $$\mathscr {C}_{d,p}$$ defined as the smallest constant C such that $$\Vert P\Vert _\infty ^p \le C\Vert P\Vert _p^p$$ holds for every polynomial P of degree d , where we consider the $$L^p$$ norm on the unit circle. We conjecture that $$\mathscr {C}_{d,p} \le dp/2+1$$ 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 $$p \ge 6.8.$$