  The spinless relativistic Coulomb problem is the bound-state problem for the
spinless Salpeter equation (a standard approximation to the Bethe--Salpeter
formalism as well as the most simple generalization of the nonrelativistic
Schr\"odinger formalism towards incorporation of relativistic effects) with the
Coulomb interaction potential (the static limit of the exchange of some
massless bosons, as present in unbroken gauge theories). The nonlocal nature of
the Hamiltonian encountered here, however, renders extremely difficult to
obtain rigorous analytic statements on the corresponding solutions. In view of
this rather unsatisfactory state of affairs, we derive (sets of) analytic upper
bounds on the involved energy eigenvalues.
