Inequalities for sine and cosine polynomials

In this paper, we prove that, letting lambda be a real number, (i) lambda Sigma(n)(k=1) (-1)(k) sin(kx) <= Sigma(n)(k=1) sin(kx)/k is valid for all n >= 1 and x is an element of [0, pi] if and only if lambda is an element of [0, 2]. This extends the classical Fejer-Jackson inequality which states that (i) holds for lambda = 0. An application of (i) reveals if a > 0 and b are real numbers, then (ii) 41/96 + Sigma(n )(k=1)cos(kx)/k+1 >= a(cos(x) + b)(2) holds for all n >= 2 and x is an element of [0, pi] if and only if a <= 2/75 and b = 3/8. This refines a result of Koumandos (2001) who proved that the expression on the left-hand side of (ii) is nonnegative for all n >= 2 and x is an element of [0,pi]. The cosine polynomial in (ii) was first studied by Rogosinski and Szego in 1928.