  The Hamiltonian of the spinless relativistic Coulomb problem combines the
standard Coulomb interaction potential with the square-root operator of
relativistic kinematics. This Hamiltonian is known to be bounded from below up
to some well-defined critical coupling constant. At this critical coupling
constant, however, the differences between all analytically obtainable upper
bounds on the corresponding energy eigenvalues and their numerically determined
(approximate) values take their maxima. In view of this, an analytical
derivation of (not so bad) upper bounds on the lowest-lying energy levels at
the critical coupling constant is presented.
