  A weaker form of the multiplicity conjecture of Herzog, Huneke, and
Srinivasan is proven for two classes of monomial ideals: quadratic monomial
ideals and squarefree monomial ideals with sufficiently many variables relative
to the Krull dimension. It is also shown that tensor products, as well as
Stanley-Reisner ideals of certain unions, satisfy the multiplicity conjecture
if all the components do. Conditions under which the bounds are achieved are
also studied.
