Bounding reduction number and the Hilbert coefficients of filtration

Let $(A,\m)$ be a Cohen-Macaulay local ring of dimension $d\geq 3$, $I$ an $\m$-primary ideal and $\mathcal{I}=\{I_n\}_{n\geq 0}$ an $I$-admissible filtration. We establish bounds for the third Hilbert coefficient: (i) $e_3(\mathcal{I})\leq e_2(\mathcal{I})(e_2(\mathcal{I})-1)$ and (ii) $e_3(I)\leq e_2(I)(e_2(I)-e_1(I)+e_0(I)-\ell(A/I))$ if $I$ is an integrally closed ideal. Further, assume the respective boundary cases along with the vanishing of $e_i(\mathcal{I})$ for $4\leq i\leq d$. Then we show that the associated graded ring of the Ratliff-Rush filtration of $\mathcal{I}$ is almost Cohen-Macaulay, Rossi's bound for the reduction number $r_J(I)$ of $I$ holds true and the reduction number of Ratliff-Rush filtration of $\mathcal{I}$ is bounded above by $r_J(\I).$ In addition, if $\wt{I^{r_J(I)}}=I^{r_J(I)}$, then we prove that $\reg G_I(A)=r_J(I)$ and a bound on the stability index of Ratliff-Rush filtration is obtained. We also do a parallel discussion on the \textquotedblleft good behaviour of the Ratliff-Rush filtration with respect to superficial sequence''.