Sharp upper bound for the point-evaluation constant on the unit circle

Establish that for every integer d ≥ 1 and every real p ≥ 2, the minimal constant \mathscr{C}_{d,p}, defined as the smallest C such that ||P||_∞^p ≤ C ||P||_p^p for all complex polynomials P of degree at most d on the unit circle (with the L^p norm on the unit circle), satisfies the inequality \mathscr{C}_{d,p} ≤ dp/2 + 1.

Background

The paper studies the norm of point evaluation at z = 1 for polynomials of degree at most d on the unit circle with respect to the L^p norm. The associated extremal constant \mathscr{C}{d,p} is defined by the inequality ||P||∞^p ≤ C ||P||_p^p holding for all such polynomials.

The authors conjecture a sharp bound \mathscr{C}_{d,p} ≤ dp/2 + 1 for all p ≥ 2 and all d. They verify this conjecture for d ≤ 4 when 2 ≤ p ≤ 4 (and hence for all p ≥ 2 by a power trick for these d), and also show that the bound holds for all d when p ≥ 6.8 via a separate large-p estimate.

This problem connects to analogous bounds in the Paley–Wiener space via a limit relation: lim_{d→∞} (\mathscr{C}_{d,p}/d) = \mathscr{C}_p, so proving the conjecture would imply \mathscr{C}_p ≤ p/2 in that setting.