Certain subclass of analytic functions defined by means of differential subordination
For \alpha\in(\pi, \pi] let \mathcal{R}_\alpha(\phi) denote the class of all normalized analytic functions in the open unit disk \mathbb{U} satisfying the following differential subordination: $f'(z)\frac{1}{2}\left(1+e^{i\alpha}\right)zf'(z)\prec\phi(z)\qquad (z\in\mathbb{U} where the function \phi(z) is analytic in the open unit disk \mathbb{U} such that \phi(0)=1$ In this paper, various integral and convolution characterizations, coefficient estimates and differential subordination results for functions belonging to the class \mathcal{R}_\alpha(\phi) are investigated. The Fekete Szeg\"{o} coefficient functional associated with the $k$th root transform [f(z^k){1/k} of functions in \mathcal{R}_\alpha(\phi) is obtained. A similar problem for a corresponding class \mathcal{R}_{\Sigma;\alpha}\phi) of bi univalent functions is also considered. Connections with previous known results are pointed out.