  We report on non-perturbative bounds for structure constants on N=4 SYM. Such
bounds are obtained by applying the conformal bootstrap recently extended to
superconformal theories. We compare our results with interpolating functions
suitably restricted by the S-duality of the theory. Within numerical errors,
these interpolations support the conjecture that the bounds found in this paper
are saturated at duality invariant values of the coupling. This extends recent
conjectures for the anomalous dimension of leading twist operators.
