Let denote by S(φ) the extremal operator defined by the compression of the unilateral shift S to the model subspace H(φ) = H 2 ⊖ φ H 2 as the following S(φ)f (z) = P (zf (z)), where P denotes the orthogonal projection from H 2 onto H(φ) and φ is an inner function on the unit disc. In this mathematical notes, we give an explicit formula of the numerical radius of S(φ) in the particular case where φ 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 [16].

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