  We consider nearly-flat-band Hubbard models of a ferromagnet, that is the
models that are weak perturbations of those flat-band Hubbard models whose
ground state is ferromagnetic for any nonzero strength $U$ of the Hubbard
repulsion. In contrast to the flat-band case, in the nearly-flat-band case the
ground state, being paramagnetic for $U$ in a vicinity of zero, turns into a
ferromagnetic one only if $U$ exceeds some nonzero threshold value $U_{th}$. We
address the question whether $U_{th}$ of the considered models is in a physical
range, therefore we attempt at obtaining possibly good estimates of the
threshold value $U_{th}$. A rigorous method proposed by Tasaki is extended and
the resulting estimates are compared with small-system, finite-size scaling
results obtained for open- and periodic-boundary conditions. Contrary to
suggestions in literature, we find the latter conditions particularly useful
for our task.
