  We present the first analysis of harmonic generation data where the full
potential of the generalized nonlinear Kramers-Kronig (K-K) relations and sum
rules is exploited. We consider two published sets of wide spectral range
experimental data of third harmonic generation susceptibility on different
polymers, the polysilane (frequency range: $0.4-2.4$ $eV$) and the
polythiophene (frequency range: $0.5-2.0$ $eV$). We show that, independent
truncated dispersion relations connect the real and imaginary part of the
moments of the third harmonic generation susceptibility
$\omega^{2\alpha}\chi^{(3)}(3\omega; \omega, \omega ,\omega)$, with $\alpha$
ranging from 0 to 3, in agreement with the theory, while there is no
convergence if we choose $\alpha=4$. We repeat the same analysis for
$\omega^{2\alpha}[\chi^{(3)}(3\omega; \omega, \omega ,\omega)]^{2}$ and show
that a larger number of independent K-K relations connect the real and the
imaginary part of the function under examination. We also compute the sum rules
for the suitable moments of the real and imaginary parts, and observe that only
considering higher powers of the susceptibility the correct vanishing sum rules
are more precisely obeyed. Sum rules providing explicit information about
structural properties of the material seem to require wider spectral range.
These constraints are expected to hold for any material and provide fundamental
tests of self-consistency that any experimental or model generated data have to
obey.
