Second Hankel Determinant for Certain Subclasses of Bi-starlike Functions Defined by Differential Operators

In this paper, we obtain  upper bounds of the initial Taylor-Maclaurin coefficients \left\vert a_{۲}\right\vert , \left\vert a_{۳}\right\vert and \left\vert a_{۴}\right\vert and of the Fekete-Szegö functional \left\vert a_{۳}-\eta a_{۲}^{۲}\right\vert for certain subclasses of analytic and bi-starlike functions \mathcal{S}_{\sigma }^{\ast }(\beta,\theta ,n,m) in the open unit disk. We have also obtained an upper bound of the functional \left\vert a_{۲}a_{۴}-a_{۳}^{۲}\right\vert for the functions in the class \mathcal{S}_{\sigma }^{\ast }(\beta ,\theta ,n,m). Moreover, several interesting applications of the results presented here are also discussed.

Keywords:

Analytic functions , Univalent functions , Bi-univalent functions , Bi-starlike functions , Subordination between analytic functions , Hankel determinant