  We explore the space of meromorphic amplitudes with extra constraints coming
from the shape of the leading Regge trajectory. This information comes in two
guises: it bounds the maximal spin of exchanged particles of a given mass; it
leads to sum rules obeyed by the discontinuity of the amplitude, which express
the softness of scattering at high energies. We assume that the leading Regge
trajectory is linear, and we derive bounds on the low-energy Wilson
coefficients using the dual and primal approaches. For the graviton-graviton
scattering in four dimensions, the maximal spin constraint leads to slightly
more stringent bounds than those that follow from general constraints of
analyticity, crossing, and unitarity. The exponential softness at high energies
is manifest in our primal approach and is not used in our implementation of the
dual approach. Nevertheless, we observe the agreement between the bounds
obtained from both. We conclude that high-energy superpolynomial softness does
not leave an obvious imprint on the low-energy observables. We exhibit a
unitary three-parameter deformation of the Veneziano amplitude for the open
string case. It has a novel, exponentially soft behavior at high energies and
fixed angles. We generalize the previous analysis of this regime and present a
stringy version of the lower bound on high-energy, fixed-angle scattering by
Cerulus and Martin.
