On the asymptotic solution of one extremal problem related to nonnegative trigonometric polynomials

For every real number $\gamma\ge1$ we denote by $K^\downarrow(\gamma)$ the least possible value of the constant term of an even nonnegative trigonometric polynomial with monotone coefficients such that all its coefficients, save for the constant term, are not less than $1$ and the sum of these coefficients equals $\gamma$. In this paper, the asymptotic estimate of $K^\downarrow(\gamma)$ is found and some extremal problems on the minimum of the constant term of an even nonnegative trigonometric polynomial are studied.