meromorphic bi univalent functions

In this paper, we investigate an interesting subclass of univalent functions. Also, we introduce a new subclass of meromorphic bi-univalent functions. We obtain the estimates on the initial Taylor-Maclurin Coefficients for functions in the interesting subclass of meromorphically bi-univalent functions defined on Δ={z∈C: 1<|z|<∞}Δ={z∈C: 1<|z|<∞}.

‎In the present paper‎, ‎we define two subclasses $Sigma(lambda‎, ‎alpha‎, ‎beta)$‎, ‎$Sigma_{mathcal C}(alpha‎, ‎beta)$ of meromorphic univalent functions and subclass $Sigma_{{mathcal B},{mathcal C}}(alpha‎, ‎beta‎, ‎lambda)$ of meromorphic bi-univalent functions‎. ‎Furthermore‎, ‎we obtain estimates on the general coefficients $|b_n|~(n ge1)$ for functions in the subclasses $Sigma(lambda‎, ‎alpha‎, ‎beta)$‎, ‎$Sigma_{mathcal C}(alpha‎, ‎beta)$ and estimates for the early coefficients of functions in subclass $Sigma_{{mathcal B},{mathcal C}}(alpha‎, ‎beta‎, ‎lambda)$ by associated subordination‎. ‎The results obtained in this paper would generalize and improve those in related works of several earlier authors‎.

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