  We analyze the (discrete) spectrum of the semirelativistic
``spinless-Salpeter'' Hamiltonian H = \beta \sqrt{m^2 + p^2} + V(r), beta > 0,
where V(r) represents an attractive, spherically symmetric potential in three
dimensions. In order to locate the eigenvalues of H, we extend the ``envelope
theory,'' originally formulated only for nonrelativistic Schroedinger
operators, to the case of Hamiltonians H involving the relativistic
kinetic-energy operator. If V(r) is a convex transformation of the Coulomb
potential -1/r and a concave transformation of the harmonic-oscillator
potential r^2, both upper and lower bounds on the discrete eigenvalues of H can
be constructed, which may all be expressed in the form E = min_{r>0} [ \beta
\sqrt{m^2 + P^2/r^2} + V(r) ] for suitable values of the numbers P here
provided. At the critical point, the relative growth to the Coulomb potential
h(r) = -1/r must be bounded by dV/dh < 2 \beta/\pi.
