  We continue our study of geometric analysis on (possibly non-reversible)
Finsler manifolds, based on the Bochner inequality established by the author
and Sturm. Following the approach of the $\Gamma$-calculus a la Bakry et al, we
show the dimensional versions of the Poincare--Lichnerowicz inequality, the
logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible
case, these inequalities were obtained by Cavalletti--Mondino in the framework
of curvature-dimension condition by means of the localization method. We show
that the same (sharp) estimates hold also for non-reversible metrics.
