Bounds for the reduction number of primary ideal in dimension three

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\ge 3$ and $I$ an $\mathfrak{m}$-primary ideal of $R$. Let $r_J(I)$ be the reduction number of $I$ with respect to a minimal reduction $J$ of $I$. Suppose depth $G(I)\ge d-3$. We prove that $r_J(I)\le e_1(I)-e_0(I)+\lambda (R\mathrm{/}I)+1+(e_2(I)-1)e_2(I)-e_3(I)$, where $e_i(I)$ are Hilbert coefficients. Suppose $d=3$ and depth $G(I^t)>0$ for some $t\ge 1$. Then we prove that $r_J(I)\le e_1(I)-e_0(I)+\lambda (R\mathrm{/}I)+t$.