  We here comment on a series of recent papers by Igi and Ishida[K. Igi and M.
Ishida, Phys. Lett B 622, 286 (2005)] and Block and Halzen[M. M. Block and F.
Halzen, Phys. Rev D 72, 036006 (2005)] that fit high energy $pp$ and $\bar pp$
cross section and $\rho$-value data, where $\rho$ is the ratio of the real to
the imaginary portion of the forward scattering amplitude. These authors used
Finite Energy Sum Rules and analyticity consistency conditions, respectively,
to constrain the asymptotic behavior of hadron cross sections by anchoring
their high energy asymptotic amplitudes--even under crossing--to low energy
experimental data. Using analyticity, we here show that i) the two apparently
very different approaches are in fact equivalent, ii) that these analyticity
constraints can be extended to give new constraints, and iii) that these
constraints can be extended to crossing odd amplitudes. We also apply these
extensions to photoproduction. A new interpretation of duality is given.
