Sharp inequalities for trigonometric sums

We prove the following two theorems: (I) Let n greater than or equal to I be a (fixed) integer. Then we have for theta is an element of (0, pi): Sigma(k=0)(n) cos(ktheta)/k less than or equal to -log(sin(theta/2)) + pi-theta/2 + sigma(n), with the best possible constant sigma(n) = Sigma(k=1)(n)(-1)(k)/k. (II) For even integers n greater than or equal to 2 and for theta is an element of (0, pi) we have Sigma(k=1)(n) sin(ktheta)/k less than or equal to alpha(pi-theta), with the best possible constant alpha = 0.66395.... Our results refine inequalities due to C. Hylten-Cavallius [11] and P. Turan [23], respectively.