An Inequality for Cosine Polynomials

We prove that if a0, a1, . . . , an are real numbers, then p1 n p1 n H02a0q k s1 ak cos kx. dxH02q k s1 cos kx. dx 1 i G min anq1yk , / 1FiFnq1 iks1 with equality holding if and only if a0sa1s ??? sanG0. This inequality refines a result of L. Fejes.