  In this article, we prove that the Buchsbaum-Rim function
$\ell_A(\S_{\nu+1}(F)/N^{\nu+1})$ of a parameter module $N$ in $F$ is bounded
above by $e(F/N) \binom{\nu+d+r-1}{d+r-1}$ for every integer $\nu \geq 0$.
Moreover, it turns out that the base ring $A$ is Cohen-Macaulay once the
equality holds for some integer $\nu$. As a direct consequence, we observe that
the first Buchsbaum-Rim coefficient $e_1(F/N)$ of a parameter module $N$ is
always non-positive.
