  In this paper, we introduce and investigate a new subclass of the function
class $\Sigma$ of bi-univalent functions defined in the open unit disk, which
are associated with the S\u{a}l\u{a}gean type $q-$ difference operator and
satisfy some subordination conditions. Furthermore, we find estimates on the
Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in the new
subclass introduced here. Several (known or new) consequences of the results
are also pointed out. Further we obtain Fekete-Szeg$\ddot{o}$ inequality for
the new function class.
