  We study multi-partite entanglement, the generation of EPR states and quantum
steering in a three-mode optomechanical system composed of an atomic ensemble
located inside a single-mode cavity with a movable mirror. The cavity mode is
driven by a short laser pulse, has a nonlinear parametric-type interaction with
the mirror and a linear beamsplitter-type interaction with the atomic ensemble.
There is no direct interaction of the mirror with the atomic ensemble. A
threshold effect for the dynamics of the system is found, above which the
system works as an amplifier and below which as an attenuator of the output
fields. The threshold is determined by the ratio of the coupling strengths of
the cavity mode to the mirror and to the atomic ensemble. It is shown that
above the threshold the system effectively behaves as a two-mode system in
which a perfect bipartite EPR state can be generated, while it is impossible
below the threshold. Furthermore, a fully inseparable tripartite entanglement
and even further a genuine tripartite entanglement can be produced above and
below the threshold. In addition, we consider quantum steering and examine the
monogamy relations that quantify the amount of bipartite steering that can be
shared between different modes. It is found that the mirror is more capable for
steering of entanglement than the cavity mode. The two way steering is found
between the mirror and the atomic ensemble despite the fact that they are not
directly coupled to each other, while it is impossible between the output of
cavity mode and the ensemble which are directly coupled to each other.
