Lower bounds for the merit factors of trigonometric polynomials from Littlewood classes

With the notation K = R(mod 2π), ||p||Lλ(K) = (∫K|p(t)|λ dt)1/λ and Mλ(p) = (1/2π ∫K|p(t)|λ dt)1/λ we prove the following result.Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with real coefficients that satisfies ||p||L2(K) ≤ An1/2 and ||p'||L2(K) ≥ Bn3/2. Then M4(p) - M2(p) ≥ eM2(p) with e = (1/111)(B/A)12. We also prove that M∞(1+2p)- M2(1 + 2p)≥(√4/3-1)M2(1+2p) and M2(p) - M1(p) ≥ 10-31M2(p) for every p ∈ An, where An denotes the collection of all trigonometric polynomials of the form p(t) = pn(t) = αj aj cos(jt + αj), aj= ±1, αj ∈ R.