Initial Coefficient estimates for a classes of m-fold symmetric bi-univalent functions involving Mittag-Leffler function

Abbas Kareem Wanas, Huo Tang

The main object of the present paper is to use Mittag-Leffler function to introduce and study two new classes \[ \mathcal{R}_{\Sigma_{m}}(\gamma,ambda,\eta,ẹlta,au;lpha)\quadext{and}\quad \mathcal{R}_{\Sigma_{m}}^{*}(\gamma,ambda,\eta,ẹlta,au;\beta) \] of $\Sigma_{m}$ consisting of analytic and $m$-fold symmetric bi-univalent functions defined in the open unit disk $U$. Also, we determine the estimates on the initial coefficients $\left| a_{m+1}\right|$ and $\left| a_{2m+1}\right|$ for functions in each of these new classes. Furthermore, we indicate certain special cases for our results.