Sharp estimates for various trigonometric sums (Q2907398)

The authors present some deep results pertaining to classical analysis. Namely, they prove sharp inequalities for finite trigonometric sums similar to the inequalities of Fejer-Jackson and Young. One example is the sharp inequality NEWLINE\[NEWLINE{-\sqrt{3}\over 18}\leq \sum^n_{k=1} {\cos((2k- 1)x)\over k}NEWLINE\]NEWLINE true for \(n=1,2,\dots\) and every \({-\pi\over 2}<x<{\pi\over 2}\). The paper contains many other interesting inequalities and evaluations for trigonometric sums and sums with binomial coefficients.