### 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.

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