On extremal properties of the nonnegative trigonometric polynomials

Let $C^+_n(a)$, ($a\geq 0$, $n\geq 1$) be the set of nonnegative trigonometric polynomials $f(t)=\sum^n_{k=0}a_k\cos kt$ with $a_0=1$, $a_1=a$, $a_k\geq 0(k=2,\dots,n)$ The function
$$ u_n(a)=\inf\biggl\{f(0)=\sum^n_{k=0}a_k:f\in C^+_n(a)\biggr\} $$
on the segment $[0,A(n)]$, $A(n)=2\cos\frac{\pi}{n+2}$, has been studied. Values of the $u_n(a)$ for the close to $A(n)$ arguments a have been obtained. The results given in the present article have been applied to the problem of Ch.-J. Vallé Poussin and E. Landau that cropped up in the course of their investigation on the prime number theory.