Sub- and superadditive properties of Fejér's sine polynomial

Abstract

Let Sn(x) = ∑k=1n (sin(kx))/k be Fejér's sine polynomial. We prove the following statements. (i) The inequality (Sn (x + y))α (x + y)β ≤ (Sn (x))α xβ + (Sn (y))αybeta

(n ∈ ℕ

α, β ∈ ℝ) holds for all x, y ∈ (0, π) with x + y < π if and only if α ≥ 0 and α + β ≤ 1. (ii) The converse of the above inequality is valid for all x, y ∈ (0, π) with x + y < π if and only if α ≤ 0 and α + β ≥ 1. (iii) For all n ∈ ℕ and x, y ∈ [0, π] we have 0 ≤ Sn (x) + S n (y) - Sn (x + y) ≤ 3/2√3. Both bounds are best possible. © 2006 London Mathematical Society.