  Using the $\alpha$-T$_3$ model, we carried out analytical and numerical
calculations for the static and dynamic polarization functions in the presence
of a perpendicular magnetic field. These results were employed to determine the
longitudinal dielectric function and the magnetoplasmon dispersion relation.
The magnetic field splits the continuous valence, conduction and flat energy
subband into discrete Landau levels which present significant effects on the
polarization function and magnetoplasmon dispersion. We present results for a
doped layer in the integer quantum Hall regime for fixed hopping parameter
$\alpha$ and various magnetic fields as well as chosen magnetic field and
different $\alpha$ in the random phase approximation.
