Small perturbations in generalized Cohen-Macaulay local rings

Let $(R,\mathfrak{m})$ be a generalized Cohen-Macaulay local ring of dimension $d$, and $f_1,\dots ,f_r$ a part of system of parameters of $R$. In this paper we give explicit numbers $N$ such that the lengths of all lower local cohomology modules and the Hilbert function of $R\mathrm{/}(f_1,\dots ,f_r)$ are preserved when we perturbs the sequence $f_1,\dots ,f_r$ by $\varepsilon_1, \ldots, \varepsilon_r \in {\frak m}^N$. The second assertion extends a previous result of Srinivas and Trivedi for generalized Cohen-Macaulay rings.