On the numerical radius of the truncated adjoint Shift

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.