New subclasses of meromorphic bi-univalent functions by associated with ‎subordinate

Document Type : Research Paper


1 Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran

2 Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-36155, Shahrood, Iran



‎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.