  Matter and quasi-matter bounce scenarios are studied for an F(R) gravity
model with holonomy corrections and a Lagrange multiplier, with a scale factor
$a(t) = \left(a_0t^2 + 1 \right)^n$, where the Hubble parameter squared has a
linear and a quadratic dependence on the effective energy density. Provided $n
< 1/2$, it is shown that the primordial curvature perturbations are generated
deeply into the contracting era, at large negative time, which makes the
low-curvature limit a good approximation for calculating the perturbation power
spectrum. Moreover, it is shown that, for $n$ within this range, the obtained
cosmological quantities are fully compatible with the Planck constraints, and
that the "low curvature limit" comes as a viable approximation to calculate the
power spectra of both scalar and tensor perturbations. Using reconstruction
techniques for F(R) gravity with the Lagrange multiplier, the precise form of
the effective F(R) gravity is found, from which one determines the power
spectra of scalar and tensor perturbations in such bouncing scenario.
Correspondingly, the spectral index for curvature perturbations and the tensor
to scalar ratio are obtained, and these values are successfully confronted with
the latest Planck observations. Further, it is shown that both the weak and the
null energy conditions are satisfied, thanks to the holonomy corrections
performed in the theory--which are then proven to be necessary for achieving
this goal. In fact, when approaching the bouncing era, the holonomy corrections
become significant and play a crucial role in order to restore the energy
conditions. Summing up, a cosmological bouncing scenario with the scale factor
above and fulfilling the energy conditions can be adequately described by the
F(R) model with a Lagrange multiplier and holonomy corrections, which prove to
be very important.
