  The rise of the total, elastic and inelastic hadronic cross sections at high
energies is investigated by means of an analytical parametrization, with the
exponent of the leading logarithm contribution as a free fit parameter. Using
derivative dispersion relations with one subtraction, two different fits to
proton-proton and antiproton-proton total cross section and rho parameter data
are developed, reproducing well the experimental information in the energy
region 5 GeV - 7 TeV. The parametrization for the total cross sections is then
extended to fit the elastic (integrated) cross section data in the same energy
region, with satisfactory results. From these empirical results we extract the
energy dependence of several physical quantities: inelastic cross section,
ratios elastic/total, inelastic/total cross sections, ratio
total-cross-section/elastic-slope, elastic slope and optical point. All data,
fitted and predicted, are quite well described. We find a statistically
consistent solution indicating: (1) an increase of the hadronic cross sections
with the energy faster than the log-squared bound by Froissart and Martin; (2)
asymptotic limits 1/3 and 2/3 for the ratios elastic/total and inelastic/total
cross sections, respectively, a result in agreement with unitarity. These
indications corroborate recent theoretical arguments by Ya. I. Azimov on the
rise of the total cross section.
