  It is a longstanding desire of cosmologists, and particle physicists as well,
to connect inflation to low energy physics, culminating, for instance, in what
is known as Higgs inflation. The condition for the standard Higgs boson playing
the role of the inflaton, and driving successfully inflation, is that it
couples nonminimally with gravity. Nevertheless, cosmological constraints
impose that the nonminimal coupling be large. This causes the loss of
perturbative unitarity in a scale of energy far below the Planck one. Our aim
in this work is to point out that inflaton potential containing a particular
type of trilinear coupling involving the inflaton may circumvent this problem
by realizing Higgs inflation with tiny nonminimal coupling of the inflaton with
gravity. We then develop the idea within a toy model and implement it in the
inverse type-II seesaw mechanism for small neutrinos masses.
