Geodesic
An inequality for the coefficients of a cosine polynomial
Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 3, pp. 427-428

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We prove: If $$ \frac 12+\sum_{k=1}^{n}a_k(n)\cos (kx)\geq 0 \text{ for all } x\in [0,2\pi ), $$ then $$ 1-a_k(n)\geq \frac 12 \frac{k^2}{n^2} \text{ for } k=1,\dots ,n. $$ The constant $1/2$ is the best possible.
Classification : 26D05, 42A05
Keywords: cosine polynomials; inequalities 
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     title = {An inequality for the coefficients of a cosine polynomial},
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Alzer, Horst. An inequality for the coefficients of a cosine polynomial. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 3, pp. 427-428. http://geodesic.mathdoc.fr/item/CMUC_1995__36_3_a2/