In this paper, we obtain  upper bounds of the initial Taylor-Maclaurin coefficients $\left\vert a_{2}\right\vert ,$ $\left\vert a_{3}\right\vert $ and $\left\vert a_{4}\right\vert $ and of the Fekete-Szegö functional $\left\vert a_{3}-\eta a_{2}^{2}\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_{2}a_{4}-a_{3}^{2}\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.