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 \(p \ge 6.8.\)
