Abstract. The inequality $\sum_{k=1}^n (n-k+a)(n-k+b) \sin(kx) \cos(ky)>0$ $(a,b\in\opr )$ is proved to hold for all $n\in\opn $ and $x,y\in\opr $ with $0<x+y< \pi $, $0<x-y< \pi $ if and only if $0< ab\leq a+b+1$. This extends a theorem of Turán, who showed that our inequality is valid for $a=1$, $b=2$, $y=0$.

DOI: 10.14232/actasm-013-562-6

AMS Subject Classification (1991): 26D05, 42A05

Keyword(s): trigonometric sums, inequalities, sine integral

Received September 12, 2013. (Registered under 62/2013.)