Constant-roll Inflation in $F(R)$ Gravity
  We propose the study of constant-roll inflation in $F(R)$ gravity. We use two
different approaches, one that relates an $F(R)$ gravity to well known scalar
models of constant-roll and a second that examines directly the constant-roll
condition in $F(R)$ gravity. With regards to the first approach, by using well
known techniques, we find the $F(R)$ gravity which realizes a given
constant-roll evolution in the scalar-tensor theory. We also perform a
conformal transformation in the resulting $F(R)$ gravity and we find the
Einstein frame counterpart theory. As we demonstrate, the resulting scalar
potential is different in comparison to the original scalar constant-roll case,
and the same applies for the corresponding observational indices. Moreover, we
discuss how cosmological evolutions that can realize constant-roll to
constant-roll eras transitions in the scalar-tensor description, can be
realized by vacuum $F(R)$ gravity. With regards to the second approach, we
examine directly the effects of the constant-roll condition on the inflationary
dynamics of vacuum $F(R)$ gravity. We present in detail the formalism of
constant-roll $F(R)$ gravity inflationary dynamics and we discuss how the
inflationary indices become in this case. We use two well known $F(R)$
gravities in order to illustrate our findings, the $R^2$ model and a power-law
$F(R)$ gravity in vacuum. As we demonstrate, in both cases the parameter space
is enlarged in comparison to the slow-roll counterparts of the models, and in
effect, the models can also be compatible with the observational data. Finally,
we briefly address the graceful exit issue.
