On a Remez-type inequality for trigonometric polynomials

We obtain a Remez-type inequality for a trigonometric polynomial QnQn of degree at most nn with real coefficients ‖Qn‖C((−π,π]≤(1/2)(2/sin(λ/4))2n‖Qn‖C(E),λ∈(0,2π], where E⊆(−π,π]E⊆(−π,π] is a measurable set with |E|≥λ|E|≥λ. This estimate is asymptotically sharp as λ→0+λ→0+, that is, for the best constant Cn,R(λ)Cn,R(λ) in this inequality, Cn,R(λ)=(1/2)(8/λ)2n(1+o(1))Cn,R(λ)=(1/2)(8/λ)2n(1+o(1)). We also extend this result to polynomials with complex coefficients.