  We show that for the straightforward quantized relativistic Coulomb
Hamiltonian of a two-dimensional atom -- or the corresponding magnetic quantum
dot -- the maximal number of electrons does not exceed twice the nuclear
charge. It result is then generalized to the presence of external magnetic
fields and atomic Hamiltonians. This is based on the positivity of $$|\bx|
T(\bp) + T(\bp) |\bx| $$ which -- in two dimensions -- is false for the
non-relativistic case $T(\bp) = \bp^2$, but is proven in this paper for $T(\bp)
= |\bp|$, i.e., the ultra-relativistic kinetic energy.
