Motivated by q-calculus, we define a new family of $\Sigma$, which is the family of bi-univalent analytic functions in the open unit disc U that is related to the Einstein function $E\left(z\right)$. We establish estimates for the first two Taylor–Maclaurin coefficients $|a_2|$, $|a_3|$, and the Fekete–Szegö inequality $\left|a_3-\mu a_2^2\right|$ for the functions that belong to these families.

In this paper, two bounded bi-univalent function subclasses were defined by using Salagean q-differential operator. The functions are defined in the open unit disc of complex plane. The main purpose is to determine some estimations on the initial Maclaurin coefficients for functions in these subclasses. Finally, the Fekete-Szegö inequalities for these are also obtained.

A novel family of bi-univalent holomorphic functions is introduced by the use of the Lindelöf principle. The upper bound of the second Hankel determinant, $H_{2,2}\left(\chi \right)$, is evaluated. Furthermore, specific results are obtained as special cases of the main conclusion. These cases coincide with certain recently obtained results and improve or enhance specific ones.

Motivated by q-calculus, subordination principle, and the second Einstein function, we define two families of bi-univalent analytic functions on the open unit disc of the complex plane. We deduce estimates for the first two Maclaurins coefficients and the Fekete-Sezg functional inequalities for the functions that belong to these families of functions.

A new class ${\mathcal{B}}_{\mathsf{\Sigma }}^{\lambda }(\gamma ,\kappa )$ of bi-starlike $\lambda$-pseudo functions related to the second Einstein function is presented in this paper. $c_2$ and $c_3$ indicate the initial Taylor coefficients of $\varphi \in {\mathcal{B}}_{\mathsf{\Sigma }}^{\lambda }(\gamma ,\kappa ),$ and the bounds for $|c_2|$ and $|c_3|$ are obtained. Additionally, for $\varphi \in {\mathcal{B}}_{\mathsf{\Sigma }}^{\lambda }(\gamma ,\kappa )$, we calculate the Fekete–Szegö functional.