Some extremal properties of positive trigonometric polynomials

For n=8 an upper bound is given for the functional $$V_n = \mathop {\inf }\limits_{t_n } \frac{{\alpha _1 + \alpha _2 + \cdots + \alpha _n }}{{\left( {\sqrt {\alpha _1 } - \sqrt {\alpha _0 } } \right)^2 }}$$ , which is defined on the class of even, nonnegative, trigonometric polynomials \(t_n (\phi ) = \sum\nolimits_{k = 0}^n {\alpha _k } cos k\phi \) , such that α _k ? 0 (k=0, ...,n) α₁>α₀ :V _s ? 34.54461566.