  We investigate the effect of an applied constant and uniform magnetic field
in the fine-structure constant of massive and massless QED. In massive QED, it
is shown that a strong magnetic field removes the so called Landau pole and
that the fine-structure constant becomes anisotropic having different values
along and transverse to the field direction. Contrary to other results in the
literature, we find that the anisotropic fine-structure constant always
decreases with the field. We also study the effect of the running of the
coupling constant with the magnetic field on the electron mass. We find that in
both cases of massive and massless QED, the electron dynamical mass always
decreases with the magnetic field, what can be interpreted as an inverse
magnetic catalysis effect.
