  With delayed and imperfect current channel state information at the
transmitter (CSIT), namely mixed CSIT, the sum degrees-of-freedom (sum-DoF) in
the two-user multiple-input multiple-output (MIMO) broadcast channel and the
$K$-user multiple-input single-output (MISO) broadcast channel with
not-less-than-$K$ transmit antennas have been obtained. However, the case of
the three-user broadcast channel with two transmit antennas and mixed CSIT is
still unexplored. In this paper, we investigate the sum-DoF upper bound of
three-user MISO broadcast channel with two transmit antennas and mixed CSIT. By
exploiting genie-aided signaling and extremal inequalities, we derive the
sum-DoF upper bound as $(1-\alpha)3/2 + 9\alpha/4$, which is at most $12.5\%$
larger than the expected sum-DoF, given by $(1-\alpha)3/2 + 2\alpha$. This
indicates that the gap may mitigate by better bounding the imperfect current
CSIT counterpart.
