  Let $(R,\mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d\geq 2$
and $I$ an $\mathfrak m$-primary ideal. Let rd$(I)$ be the reduction number of
$I$ and n$(I)$ the postulation number. We prove that for $d=2,$ if
n$(I)=\rho(I)-1,$ then rd$(I) \leq$n$(I)+2$ and if n$(I)\neq \rho(I)-1,$ then
rd$(I)\geq$n$(I)+2.$ For $d \geq 3$, if $I$ is integrally closed, depth gr$(I)
= d-2$ and n$(I)=-(d-3).$ Then we prove that rd$(I)\geq$n$(I)+d$. Our main
result is to generalize a result of T. Marley on the relation between the
Hilbert-Samuel function and the Hilbert-Samuel polynomial by relaxing the
condition on the depth of the associated graded ring with the good behaviour of
the Ratliff-Rush filtration with respect to $I$ mod a superficial element. From
this result, it follows that for a Cohen-Macaulay ring of dimension $d\geq2$,
if $P_{I}(k)=H_{I}(k)$ for some $k \geq \rho(I)$, then $P_{I}(n)=H_{I}(n)$ for
all $n \geq k.$
