ABSTRACTL.e t T be a bounded linear operatoro f norm 1 on a Hilbert space H such that T" = 0 for some n > 2. Then its numerical radius satisfies w (T) < cos (n'n+ 1) and this bound is sharp. Moreover, if there exists a unit vector s E H such that I( TIg) l= cos (n+'+1 ) then T has a reducings ubspace of dimension n on which T is the usual n-shift. The proofs show that these facts are related to the following result of Fejer: if a trigonometric polynomial f(6) = Zk-!n+i fkeikO is positive, one has If, I < fo cos (n+); moroever, there is essentially one polynomial for which equality holds.