Geodesic
Some extremal properties of positive trigonometric polynomials
Matematičeskie zametki, Tome 22 (1977) no. 3, pp. 371-374 Cet article a éte moissonné depuis la source Math-Net.Ru

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For $n=8$ an upper bound is given for the functional $$ V_n=\inf_{t_n}\frac{a_1+a_2+\dots+a_n}{(\sqrt{a_q}-\sqrt{a_0})^2}, $$ which is defined on the class of even, nonnegative, trigonometric polynomials $t_n(\varphi)=\sum_{k=0}^na_k\cos k\varphi$, such that $a_k\ge0$ ($k=0,\dots,n$), $a_1>a_0:V_8\le34,\!54461566$. 
@article{MZM_1977_22_3_a5,
     author = {V. P. Kondrat'ev},
     title = {Some extremal properties of positive trigonometric polynomials},
     journal = {Matemati\v{c}eskie zametki},
     pages = {371--374},
     year = {1977},
     volume = {22},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_3_a5/}
}
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V. P. Kondrat'ev. Some extremal properties of positive trigonometric polynomials. Matematičeskie zametki, Tome 22 (1977) no. 3, pp. 371-374. http://geodesic.mathdoc.fr/item/MZM_1977_22_3_a5/