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. 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 $bar{p}p$ scattering, in the region 5 GeV-8 TeV. The recent tension between the TOTEM and ATLAS results at 7 TeV and mainly 8 TeV is considered in the data reductions. Our main conclusions are: (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 $gammasim 2.0pm 0.2$ (AU) and $gammasim 2.3pm 0.1$ (DDR); (3) by including the ATLAS data the fits provide $gammasim 1.9pm 0.1$ (AU) and $gammasim 2.2pm 0.2$ (DDR); (4) in the formal and practical contexts, the DDR approach is more adequate for the energy interval investigated than the AU approach. A review on the analytic results for $sigma_{tot}$ and $rho$ from the Regge-Gribov, DDR and AU approaches is presented.
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