  Let denote by $S(\phi)$ the extremal operator defined by the compression of
the unilateral shift $S$ to the model subspace $ H(\phi)=H^{2} \ominus \phi
H^{2} $ as the following $S(\phi)f(z)=P(zf(z)),$ where $P$ denotes the
orthogonal projection from the Hardy space $H^{2}$ onto $ H(\phi)$ and $\phi$
is an inner function on the unit disc. In this mathematical notes, we give an
explicit formula of the numerical radius of the truncated shift $S(\phi)$ in
the particular case where $\phi$ is a finite Blaschke product with unique zero
and an estimate on the general case. We establish also a sharpened Schwarz-Pick
operatorial inequality generalizing a U. Haagerup and P. de la Harpe result for
nilpotent operators
