On an extremal problem for polynomials with fixed mean value

Let $T_n^+$ be the set of nonnegative trigonometric polynomials $\tau_n$ of degree $n$ that are strictly positive at zero. For $0\le\alpha\le2\pi/(n+2),$ we find the minimum of the mean value of polynomial $(\cos\alpha-\cos{x})\tau_n(x)/\tau_n(0)$ over $\tau_n\in{T_n^+}$ on the period $[-\pi,\pi).$
The paper was originally published in a hard accessible collection of articles Approximation of Functions by Polynomials and Splines (The Ural Scientific Center of the Academy of Sciences of the USSR, Sverdlovsk, 1985), p. 15–22 (in Russian).