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: \[ f'(z)+frac12(1+e^{ilpha})zf''(z)rechi(z)\qquad(zı\mathbb U) \] 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-Szegö 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.