  An equivalent formulation of the von Neumann inequality states that the
backward shift $S^*$ on $\ell_{2}$ is extremal, in the sense that if $T$ is a
Hilbert space contraction, then $\|p(T)\| \leq \|p(S^*)\|$ for each polynomial
$p$. We discuss several results of the following type : if $T$ is a Hilbert
space contraction satisfying some constraints, then $S^*$ restricted to a
suitable invariant subspace is an extremal operator. Several operator radii are
used instead of the operator norm. Applications to inequalities of coefficients
of rational functions positive on the torus are given.
