Sharp inequalities for trigonometric sums

Abstract

We prove the following two theorems: (I) Let n ≥ 1 be a (fixed) integer. Then we have for θ ∈ (0, π): ∑k=1ncos(kθ)/k ≤ -log (sin (θ/2)) + π-θ/2 + σn, with the best possible constant σn = ∑k=1n (-1)k/k. (II) For even integers n ≥ 2 and for θ ∈ (0, π) we have ∑k=1n sin(kθ)/k ≤ α(π-θ), with the best possible constant α = 0.66 395.... Our results refine inequalities due to C. Hyltén-Cavallius [11] and P. Turán [23], respectively.