On a trigonometric inequality of Turán

We prove the following theorem: Let m≥2m≥2 be a given integer and let a,b,ca,b,c be real numbers. The inequalities ∑k=1nm+n−kmsin(kx)>ax2+bx+c>0 hold for all integers n≥2n≥2 and real numbers x∈(0,π)x∈(0,π) if and only if −m−1π≤a<0,b=−aπ,c=0. This refines a result due to Turán.