  Recently there are several evidences of the hadronic total cross section
sigma(tot) to be proportional to B (log s)2 consistent with the Froissart
unitarity bound. The COMPETE collaborations have further assumed sigma(tot) = B
(log s/s0)2 + Z to extend its universal rise with the common value of B and s0
for all hadronic scatterings to reduce the number of adjustable parameters. The
coefficient B was suggested to be universal in the arguments of colour glass
condensate (CGC) of QCD in recent years. There has been, however, no rigorous
proof yet based only on QCD. We attempt to investigate the value of B for pi+-
p, K+- p and pbar p,pp scatterings respectively through the search for the
simultaneous best fit to the experimental sigma(tot) and rho ratios at high
energies. The sigma(tot) at the resonance and intermediate energy regions has
also been exploited as a duality constraint based on the special form of
finite-energy sum rule(FESR). We estimate the values of B, s0 and Z
individually for pi+- p, K+- p and pbar p,pp scatterings without using the
universality hypothesis. It turs out that the values of B are mutually
consistent within one standard deviation. It has to be stressed that we cannot
obtain such a definite conclusion without the duality constraint. It is also
interesting to note that the values of Z for pi p, Kp and pbar(p)p
approximately satisfy the ratio 2:2:3 predicted by the quark model. The
obtained value of B for pbar(p)p is Bpp = 0.280 +- 0.015 mb, which predicts
sigma(tot)(pp) =108.0 +- 1.9mb and rho(pp) =0.131+- 0.0025 at the LHC energy
s^0.5 =14 TeV.
