  Let S=K[x_1,...,x_n] be a polynomial ring and R=S/I be a graded K-algebra
where I is a graded ideal in S. Herzog, Huneke and Srinivasan have conjectured
that the multiplicity of R is bounded above by a function of the maximal shifts
in the minimal graded free resolution of R over S. We prove the conjecture in
the case that codim(R)=2 which generalizes results in of Herzog, Srinivasan and
Gold. We also give a proof for the bound in the case in which I is
componentwise linear. For example, stable and squarefree stable ideals belong
to this class of ideals.
