Sharp inequalities for sine polynomials

Let F-n(x) = Sigma(n)(k=1) sin(kx)/k and C-n(x) = Sigma(n)(k=1) sin((2k - 1)x)/2k - 1. The classical inequalities 0 < F-n(x) < integral(pi)(0) sin(t)/t dt = 1.85193 ... and 0 < C-n(x) <= 1 are valid for all n >= 1 and x is an element of (0, pi). All constant bounds are sharp. We present the following refinements of the lower bound for F-n(x) and the upper bound for C-n(x). (i) Let mu >= 1. The inequality sin(x)/mu - cos(x) < F-n(x) holds for all odd n >= 1 and x is an element of (0, pi) if and only if mu >= 2. (ii) For all n >= 2 and x is an element of [0, pi], we have C-n(x) <= 1 - lambda sin(x) with the best possible constant factor lambda = 3 root 9 - 2. Moreover, we offer a companion to the inequality C-n(x) > 0. (iii) Let n >= 1. The inequality 0 <= Sigma(n)(k=1)(delta(n) - (k - 1)k) sin((2k - 1)x) holds for all x is an element of [0, pi] if and only if delta(n) >= (n(2) - 1)/2. This extends a result of Dimitrov and Merlo, who proved the inequality for the special case delta(n) = n(n + 1). The following inequality for the Chebyshev polynomials of the second kind plays a key role in our proof of (iii). (iv) Let m >= 0. For all t is an element of R, we have (m(2)(1 - t(2)) - 1) U-m(2)(t) + (m + 1)U-2m(t) <= m(m + 1). The upper bound is sharp.