  If, after primordial inflation, the universe undergoes a relatively long
reheating period, it could present a phase of matter domination supported by
the oscillating inflaton field. During this epoch, small perturbations from the
inflaton that reenter the cosmological horizon could virialize to form
\textit{inflaton} structures. If the primordial overdensities are large enough,
their associated inflaton structures could collapse to form primordial black
holes (PBHs) [L.E.Padilla, J.C.Hidalgo and K.A.Malik, Phys.Rev.D, vol.106,
p.023519, Jul 2022; hereinafter P1]. For this to happen at a considerable rate,
the primordial power spectrum should be enhanced at small scales, a feature
typically induced in single-field inflation through an ultra-slow roll phase
(produced by a nearly-inflection point in the inflationary potential). In this
article we consider two specific inflationary potentials that present this
nearly-inflection point and we look at the PBH formation rate through the
mechanism proposed in P1. We report on constraints to these two specific models
from the bounds to PBH abundances. This serves as an illustration of the
usefulness of the PBH formation mechanism proposed in P1.
