Suppose that a trigonometric polynomial
$$\tau (e^{i\theta})=\sum_{k=-N+1}^{N-1}\alpha_ke^{ik\theta},
\ \ \theta \in [0,2\pi),$$
is positive, $\alpha_{N-1}\not=0, N\ge 2.$
Then a classical matter due to Fej\'er asserts that the estimate
\Disp{|\alpha_1 |\le \alpha_0 \cos \frac{\pi}{N+1}}
for the modulus \disp{|\alpha_1|} of \disp{\alpha_1} holds
and that the equality occurs only for the polynomial 
\disp{\alpha_0 \tau_{N}(e^{i(\theta-\varphi)})},
where
\Disp{\tau_{N} (e^{i\theta})=\frac{2}{N+1} 
\left|\sum_{k=0}^{N-1}\left(\sin \frac{(k+1)\pi}{N+1}\right)
e^{ik \theta}\right|^2,\ \ \
\theta\in [0,2\pi),} and $\varphi \in [0, 2\pi).$
In this paper, we will show that the corresponding estimate
$$|\alpha_n|\le \alpha_0 \cos \frac{\pi}{\left\lceil N/n\right\rceil +1}$$
for the modulus $|\alpha_n|$ of $\alpha_n$ is true, $1\le n\le N-1,$
$\lceil N/n\rceil$ the minimum integer
not smaller than $N/n$,
and that the equality for $n=n_0$ occurs
only for the polynomial $\tau$ of the form 
$$\tau(e^{i\theta})=
\sigma (e^{i\theta})\tau_{\lceil N/n_0 \rceil} (e^{in_0(\theta -\varphi)}),
\ \ \
\theta\in [0,2\pi),$$
where $\sigma$ is a positive trigonometric polynomial and 
$\varphi \in [0, 2\pi).$
%$\tau_M$ for positive integer $M$ is:
%$$\tau_M(e^{i\theta})=\frac{2}{M+1} 
%\left|\sum_{k=0}^{M-1}\left(\sin \frac{(k+1)\pi}{M+1}\right)e^{ik \theta}
%\right|^2,\ \ \
%\theta\in [0,2\pi),$$
%and $\varphi \in [0, 2\pi),$