  Motivated by recent experimental study of antiferromagnetic property of
honeycomb compound In$_{3}$Cu$_{2}$VO$_{9}$ [Yan \textit{et al.}, PRB
\textbf{85}, 085102 (2012)], we explore possible superconductivity and its
coexistence with antiferromagnetism. We use the t-t$^\prime$-J model on the
honeycomb lattice as our starting point and employ the slave-boson mean-field
theory. In the antiferromagnetic normal state, the characteristic doping
evolution of Fermi surface shows that only one effective singe band is active,
which suggests that the potential pairing symmetry is the time-reversal
symmetry breaking $d+id$, rather than the extended $s$-wave. It is found that
this superconducting state coexists with the antiferromagnetism in a broad
doping regime, which is consistent with the numerical calculations. The local
density of states and its thermodynamic property of the superconducting state
has been studied in detail with an effective single-band picture for
understanding other physical observable such as superfluid density. The present
work may be useful in experimentally exploring possible superconductivity of
this kind of materials on the honeycomb lattice and contributes to the
understanding of the unconventional superconductivity on general
two-dimensional correlated electron systems.
