A Certain Subclass of Analytic Functions Defined by Means of Differential Subordination

Abstract

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.