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.