Let $\mathsf{\Omega }$ denote the class of functions $f\left(z\right)=z+a_2{z}^2+a_3{z}^3+\cdots$ belonging to the normalized analytic function class $\mathcal{A}$ in the open unit disk $\mathbb{U}=\left\{z:\left|z\right|<1\right\}$ , which are bi-univalent in $\mathbb{U}$ , that is, both the function f and its inverse $f^{-1}$ are univalent in $\mathbb{U}$ . In this paper, we introduce and investigate two new subclasses of the function class $\mathsf{\Omega }$ of bi-univalent functions defined in the open unit disc $\mathbb{U}$ , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

In the current study, we provide a novel qualitative new subclass of analytical and bi-univalent functions in the symmetry domain $\mathbb{U}$ defined by the use of Gegenbauer polynomials. We derive estimates for the Fekete–Szegö functional problems and the Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for the functions that belong to each of these new subclasses of the bi-univalent function classes. Some more results are revealed after concentrating on the parameters employed in our main results.

In this paper, the function classes ${\mathcal{SP}}_p(\sigma ,\nu )$ and $\mathcal{UCSP}(\sigma ,\nu )$ are investigated for the normalized Wright functions. More precisely, several sufficient and necessary conditions are provided so that the aforementioned functions are in these classes. Furthermore, several corollaries will follow from our results.