We present new accurate lower bounds on the ground state of the Hamilt
onian root p(2)+m(2)-alpha/r. From these, we obtain a lower bound on t
he maximum mass of a boson star in the semirelativistic framework, and
we compare it to a variational upper bound. We get 0.846 (Gm)(-1)<M(m
ax)<1.439(Gm)(-1), where G is Newton's constant and m the boson mass.