  A celebrated thorem of Fejer (1915) asserts that for a given positive
trigonometric polynomial
  $\sum_{j=-n+1}^{n-1}c_{j}e^{ijt}$, we have $\lvert c_{1}\lvert\leqslant
c_{0}\cos\frac{\pi}{n+1}$. A more recent inequality due to U. Haagerup and P.
de la Harpe asserts that, for any contraction $T$ such that $T^{n}=0$, for some
$n\geq2$, the inequality $\omega_{2}(T)\leqslant\cos\frac{\pi}{n+1}$ holds, and
$\omega_{2}(T)=\cos\frac{\pi}{n+1}$ when T is unitarily equivalent to the
extremal operator ${S}^{\ast}_{n}={\bbs}_{\lvert{\C}^{n}}={\bbs}_{\lvert Ker
(u_{n}(\bbs))}$ where $u_{n}(z)=z^{n}$ and $\bbs$ is the adjoint of the shift
operator on the Hilbert space of all square summable sequences. Apparently
there is no relationship between them. In this mathematical note, we show that
there is a connection between Taylor coefficients of positive rational
functions on the torus and numerical radius of the extremal operator
$\bbs(\phi)=\bbs_{\lvert Ker(\phi(\bbs))}$ for a precise inner function $\phi$.
This result completes a line of investigation begun in 2002 by C. Badea and G.
Cassier \cite{Cassier}. An upper and lower bound of the numerical radius of
$\bbs(\phi)$ are given where $\phi$ is a finite Blashke product with unique
zero.
