Buchsbaum homogeneous algebras with minimal multiplicity

In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous $k$-algebras $A$ in terms of the dimension $d$, the codimension $c$, the initial degree $q$, and the length of the local cohomology modules of $A$. Next, we introduce the notion of Buchsbaum $k$-algebras with minimal multiplicity of degree $q$, and give several characterizations for those rings. In particular, we will show that those algebras have linear free resolutions. Further, we will give many examples of those algebras.