Some extremal properties of positive trigonometric polynomials

A class $P_n$ of even positive trigonometric polynomials $t_n(\varphi)=a_0+a_1\cos\varphi+\dots+a_n\cos n\varphi$, satisfying the conditions: $a_k\geqslant0$ ($k=0,1,\dots,n$), $a_0<a_1$ is considered. The behavior of the sequence of functionals
$$ V_n=\inf_{t_n\in P_n}\frac{t_n(0)-a_0}{(\sqrt{a_1}-\sqrt{a_0})^2}, $$
is studied; two-sided estimations are given for $V_n$ and $V_\infty=\lim\limits_{n\to\infty}V_n$.