  In any consistent massive quantum field theory there are well known bounds on
scattering amplitudes at high energies. In conformal field theory there is no
scattering amplitude, but the Mellin amplitude is a well defined object
analogous to the scattering amplitude. We prove bounds at high energies on
Mellin amplitudes in conformal field theories, valid under certain technical
assumptions. Such bounds are derived by demanding the absence of spurious
singularities in position space correlators. We also conjecture a stronger
bound, based on evidence from several explicit examples.
