  Faltings' annihilator theorem is an important result in local cohomology
theory. Recently, Doustimehr and Naghipour generalized the Falitings'
annihilator theorem. They proved that if $R$ is a homomorphic image of a
Gorenstein ring, then $f_\mathfrak{a}^\mathfrak{b}(M)_n =
\lambda_\mathfrak{a}^\mathfrak{b}(M)_n$, where
$f_\mathfrak{a}^\mathfrak{b}(M)_n := \inf\{i \in \mathbb{N} \mid
\operatorname{dim}{\operatorname{Supp}(\mathfrak{b}^t H_\mathfrak{a}^i(M))}
\geq n \text{ for all } t\in \mathbb{N}\}$ and
$\lambda_\mathfrak{a}^\mathfrak{b}(M)_n := \inf\{\lambda_{\mathfrak{a}
R_\mathfrak{p}}^{\mathfrak{b} R_\mathfrak{p}}(M_\mathfrak{p}) \mid
\mathfrak{p}\in\operatorname{Spec}{R} \text{ with }
\operatorname{dim}{R/\mathfrak{p}} \geq n\}$. In this paper, we study the
relation between $f_\mathfrak{a}^\mathfrak{b}(M)_n$ and
$\lambda_\mathfrak{a}^\mathfrak{b}(M)_n$, and prove that if $R$ is an almost
Cohen-Macaulay ring, then $f_\mathfrak{a}^\mathfrak{b}(M)_n \geq
\lambda_\mathfrak{a}^\mathfrak{b}(M)_n - \operatorname{cmd}{R}$. Using this
result, we prove that if $R$ is a homomorphic image of a Cohen-Macaulay ring,
then $f_\mathfrak{a}^\mathfrak{b}(M)_n =
\lambda_\mathfrak{a}^\mathfrak{b}(M)_n$.
