A note on Ratliff-Rush filtration, reduction number and postulation number of $\mathfrak{m}$-primary ideals

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\ge 2$ 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$(I)\le$n$(I)+2$ and if n$(I)\mathrm{\ne }\rho (I)-1,$ then rd$(I)\ge$n$(I)+2.$ For $d\ge 3$, if $I$ is integrally closed, depth gr$(I)=d-2$ and n$(I)=-(d-3)\mathrm{.}$ Then we prove that rd$(I)\ge$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 $d\ge 2$, if $P_I(k)=H_I(k)$ for some $k\ge \rho (I)$, then $P_I(n)=H_I(n)$ for all $n\ge k\mathrm{.}$