Companions of the inequalities of Fejér--Jackson and Young

Summary Applications of some well-known theorems of Jackson and Young lead to the sharp inequalities -1<ⁿ_k-1Σ(cos(kx)+sin(kx))/k (n ≥1; 1<x<π) and -1/2Si(π)<ⁿ_k-1Σ(cos(kx)·sin(kx))/k (n ≥1; xЄR) We prove that the following counterpart is valid for all integers n ≥1 and real numbers xЄ (0, π): -3/2≤ⁿ_k-1Σ(cos(kx)-sin(kx))/k where the sign of equality holds if and only if n =2 and x = π /2.