In this paper, subclasses of  the function class  ∑  of analytic and bi-univalent functions associated with operator  L_q^(k,  λ)  are introduced and defined in the open unit disk △ by applying quasi-subordination. We obtain some results about the corresponding bound estimations of the coefficients  a_(2 )  and a_(3 ).

In this article, we aim to describe a new operator J_(s,a,μ)^(δ,λ) via convolution. Moreover, we aim to present a new subclass C_Σm (τ;β) related to m-fold symmetric bi-univalent functions in the open unit disk Θ={z∈C∶|z| ˂ 1 }. Finally, an estimate related to the Hankel determinant for functions in C_Σm (τ;β) are given.