In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of $n-th$$(n\ge 3)$ Taylor–Maclaurin coefficients $\left|a_n\right|$ is obtained. Furthermore, the bounds value of the first two coefficients of such functions is established.

In this article, we make use of the q-binomial theorem to introduce and study two new subclasses $\aleph (\alpha q,q)$ and $\aleph (\alpha ,q)$ of meromorphic functions in the open unit disk $\mathbb{U}$, that is, analytic functions in the punctured unit disk ${\mathbb{U}}^{\ast }=\mathbb{U}\backslash \left\{0\right\}=\{z:z\in \mathbb{C}\mathrm{and}0<\left|z\right|<1\}.$ We derive inclusion relations and investigate an integral operator that preserves functions which belong to these function classes. In addition, we establish a strict inequality involving a certain linear convolution operator which we introduce in this article. Several special cases and corollaries of our main results are also considered.