Halit Orhan - Department of Mathematics, Faculty of Science, Atatürk University, ۲۵۰۴۰, Erzurum, Turkiye.
Hava Arikan - Department of Mathematics, Faculty of Science, Atatürk University, ۲۵۰۴۰, Erzurum, Turkiye.
Murat Çağlar - Department of Mathematics, Faculty of Science, Erzurum Technical University, ۲۵۱۰۰, Erzurum, Turkiye.

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.

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