  We find estimates on the norms commutators of the form [f(x), y] in terms of
the norm of [x, y] assuming that x and y are contractions in a C*-algebra A,
with x normal and with spectrum within the domain of f. In particular we
discuss [x^2, y] and [x^(1/2), y] for 0 <=, x <=, 1. For larger values of
\delta = \|[x; y]\| we can rigorous calculate the best possible upper bound
\|[f(x), y]\| for many f. In other cases we have conducted numerical
experiments that strongly suggest that we have in many cases found the correct
formula for the best upper bound.
