In this work, we introduce and investigate a subclass Hh,p Σm (τ, γ) of analytic and bi-univalent functions when both f (z) and f −1(z) are m-fold symmetric in the open unit disk U. Moreover, we find upper bounds for the initial coefficients |am+1| and |a2m+1| for functions belonging to this subclassHh,pΣm (τ, γ). The results presented in this paper would generalize and improve those that were given in several recent works