New examples of nonnegative trigonometric polynomials with integer coefficients

In the paper it is proved that for any positive integer $n$ and any number $\lambda\geq1$ the following estimate holds:
$$ 2\lambda n^{\alpha}+\sum_{k=1}^{s}\Bigl[\lambda\left(\frac{n}{k}\right)^{\alpha}-1\Bigr]\cos(kx)>0 $$
for all $x$ and $s=0,\ldots,n$. Here the braces mean the integer part of a number, and $\alpha\in(0,1)$ is the unique root of the equation $\int_{0}^{3\pi/2}t^{-\alpha}\cos t\,dt=0$. It is proved also that for any positive integer $n$ and any numbers $q\geq2$ and $\lambda \geq sq^q$ the following estimate is true:
$$ 4\lambda n^{1/q}+\sum_{k=1}^{n}\Bigl[\lambda\Bigl(\left( \frac{n}{k}\right)^{1/q}-1\Bigr)+1\Bigl]\cos(kx)>0 $$
for all $x$. From these two main results and similar ones new estimates in some extremal problems connected with nonnegative trigonometric polynomials with integer coefficients are deduced.