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 investigate an interesting subclass of uni-valent functions. Also, we introduce a new subclass of Meromorphic bi-univalent functions. We obtain the estimates on the initial Taylor-Maclurin Coe cients for functions in the interesting subclass of meromor-phically bi-univalent functions de ned on Δ = {z∈ C: 1<|z|<∞ }.

In the present paper, we obtain some subordination and superordination results, involving the operator $T^{a}$ for functions of the form $f(z)=z^{-1}+\sum_{k=1}^{\infty}a_k z^{k}$, which are Meromorphic univalent in the punctured open unit disk these results are applied to obtain sandwich results.

In the present investigation, we use the Horadam Polynomials to establish upper bounds for the second and third coe, cients of functions belongs to a new subclass of analytic and ,-pseudo-starlike bi-univalent functions de, ned in the open unit disk U. Also, we discuss Fekete-Szeg, o problem for functions belongs to this subclass.

The purpose of the present paper is to introduce and investigate two new subclasses KΣ m(λ , γ ; α ) and K∗ Σ m (λ , γ ; β ) of Σ m consisting of analytic and m-fold symmetric bi-univalent functions defined in the open unit disk U. We obtain upper bounds for the coefficients |am+1| and |a2m+1| for functions belonging to these subclasses. Many of the well-known and new results are shown to follow as special cases of our results.

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.

In this article, the authors introduce two new subclasses of a class m-fold symmetric biunivalent functions in open unit disk. Coefficient bounds for the Taylor-Maclaurin coefficients |am+1| and |a2m+1| are obtained. Furthermore, we solve ”Fekete-Szeg” ”o” functional problems for functions in FP,m(γ, µ, ϑ) and MP,m(κ, η, ϑ). Also, several certain special improver results for the associated classes are presented.

In this paper, we introduce a newly defined  subclass $\mathcal{S}_{\Sigma}(\vartheta,\gamma,\eta;\varphi) $ of bi-univalent functions by using the Tremblay differential operator satisfying subordinate conditions in the unit disk. Moreover, we use the Faber polynomial expansion to derive bounds for the Fekete-Szego problem and first two \emph{Taylor-Maclaurin coefficients} $|a_2|$ and $|a_3|$ for functions of this class.

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.