Abstract: Forward amplitude analyses constitute an important approach in the investigation of the energy dependence of the total hadronic cross-section sigma(tot) and the rho parameter. The standard picture indicates for sigma(tot) a leading log-squared dependence at the highest c.m. energies, in accordance with the Froissart-Lukaszuk-Martin bound and as predicted by the COMPETE Collaboration in 2002. Beyond this log-squared (L2) leading dependence, other amplitude analyses have considered a log-raised-to-gamma form (L gamma), with gamma as a real free fit parameter. In this case, analytic connections with rho can be obtained either through dispersion relations (derivative forms), or asymptotic uniqueness (Phragmen-Lindeloff theorems). In this work, we present a detailed discussion on the similarities and mainly the differences between the Derivative Dispersion Relation (DDR) and Asymptotic Uniqueness (AU) approaches and results, with focus on the L gamma and L2 leading terms. We also develop new Regge-Gribov fits with updated dataset on sigma(tot) and rho from pp and (p) over barp scattering, including all available data in the region 5 GeV-8 TeV. The recent tension between the TOTEM and ATLAS results at 7 TeV and mainly at 8 TeV is discussed and considered in the data reductions. Our main conclusions are the following: (1) all fit results present agreement with the experimental data analyzed and the goodness-of-fit is slightly better in case of the DDR approach; (2) by considering only the TOTEM data at the LHC region, the fits with L gamma indicate gamma similar to 2.0 +/- 0.2 (AU approach) and gamma similar to 2.3 +/- 0.1 (DDR approach); (3) by including the ATLAS data the fits provide gamma similar to 1.9 +/- 0.1 (AU) and gamma similar to 2.2 +/- 0.2 (DDR); (4) in the formal and practical contexts, the DDR approach is more adequate for the energy interval investigated than the AU approach. A pedagogical and detailed review on the analytic results for sigma(tot) and rho from the Regge-Gribov, DDR and AU approaches is presented. Formal and practical aspects related to forward amplitude analyses are also critically discussed Ver menos