New subclasses of bi-univalent functions defined with $q$-derivative operator

In the present paper, we use the $q$-derivative operator to introduce two new and unified subclasses of bi-univalent functions in the open unit disc $E=\left\{z:\mid z \mid \lt 1\right\}$. We obtain the bounds for the initial coefficients and Fekete-Szegö inequality for these classes. Some earlier known results follow as special cases.