Rogosinski–Szegö type inequalities for trigonometric sums

We prove that the inequalities ∑k=1nsin(kx)k+1≥1384(9−137)110−6137=−0.044419686... and ∑k=1nsin(kx)+cos(kx)k+1≥−12 are valid for all real numbers x∈[0,π]x∈[0,π] and all positive integers nn. The constant lower bounds are sharp. Our theorems complement a classical result of Rogosinski and Szegö, who proved in 1928 that the inequality ∑k=1ncos(kx)k+1≥−12 holds for all x∈[0,π]x∈[0,π] and n≥1n≥1.