  Using conformal field theory (CFT) arguments we derive an infinite number of
constraints on the large spin expansion of the anomalous dimensions and
structure constants of higher spin operators. These arguments rely only on
analiticity, unitarity, crossing-symmetry and the structure of the conformal
partial wave expansion. We obtain results for both, perturbative CFT to all
order in the perturbation parameter, as well as non-perturbatively. For the
case of conformal gauge theories this provides a proof of the reciprocity
principle to all orders in perturbation theory and provides a new "reciprocity"
principle for structure constants. We argue that these results extend also to
non-conformal theories.
