  The effective Lagrangian and mass operator are calculated for planar charged
massive and massless fermions in a constant external homogeneous magnetic field
in the one-loop approximation of the 2+1 dimensional quantum electrodynamics
(QED$_{2+1}$). We obtain the renormalizable effective Lagrangian and the
fermion mass operator for a charged fermion of mass $m$ and then calculate
these quantities for the massless case. The radiative corrections to the mass
of charged massless fermion when it occupies the lowest Landau level are found
for the cases of the pure QED$_{2+1}$ as well as the so-called reduced
QED$_{3+1}$ on a 2-brane. The fermion masses were found can be generated
dynamically by an external magnetic field in the pure QED$_{2+1}$ if the
charged fermion has small bare mass $m_0$ and in the reduced QED$_{3+1}$ on a
2-brane even at $m_0=0$. The dynamical mass seems to be likely to be revealed
in monolayer graphene in the presence of constant homogeneous magnetic field
(normal to the graphene sample).
