Let 𝝨 be the class of meromorphic bi-univalent functions f of the formf(z)=z+b_0+∑ _(n=1)^∞ ▒ b_n/z^n, which are univalent (analytic and one to one) on the domain Δ ={z∈ C∶ 1.

In this paper, we introduce and investigate a subclass Hh, pS(b) of analytic and bi-univalent functions in the open unit disk U. Upper bounds for the second and third Coefficients of functions in this subclass are founded. Our results generalize and improve over the existing results in the literature.

The purpose of the present paper is to introduce two new subclasses of the function class ∑m of bi-univalent functions which both f and f-1 are m-fold symmetric analytic functions. Furthermore, we obtain estimates on the initial Coefficients for functions in each of these new subclasses. Also we explain the relation between our results with earlier known results.

In this paper, we introduce and investigate a subclass  $\mathcal{G}_{\Sigma}^{h,p}(\lambda,m,n, \alpha,\gamma)$ of bi-univalent functions in the open unit disk $\mathbb{U}$. Upper bounds for this class's second and third Coefficients of functions are found. The results, which we have presented in this paper, would generalize and improve some recent works of several earlier authors.

In the present paper, we introduce a new subclass H∑m (λ, β)of the m-fold symmetric bi-univalent functions. Also, we find the estimates of the Taylor-Maclaurin initial Coefficients |am+1|, |a2m+1| and general Coefficients |amk+1| (k ≥ 2) for functions in this new subclass. The results presented in this paper would generalize and improve some recent works of several earlier authors.

The purpose of the present paper is to introduce a class $\boldsymbol{D}% _{\lambda, \delta }^{k, \alpha }C_{0}(\beta )$ of bi-concave functions defined by a differential operator. We find estimates on the Taylor-Maclaurin Coefficients $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $ for functions in this class. Several consequences of these results are also pointed out in the form of corollaries.