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.

We study the Hadamard product features of certain subclasses of p-valent meromorphic functions defined in the punctured open-unit disc using the q-difference operator. For functions belonging to these subclasses, we obtained certain coefficient estimates and inclusion characteristics. Furthermore, linkages between the results given here and those found in previous publications are highlighted.

Over the last several years, there has been a considerable improvement in the possible methods for solving fractional-order chaotic systems; however, achieving high accuracy remains a challenge. This work proposes a new precise numerical technique for fractional-order chaotic systems. Through simulations, we obtain new types of complex and previously undiscussed dynamic behaviors.These phenomena, not recognized in prior numerical results or theoretical estimations, underscore the unique dynamics present in fractional systems. We also study the effects of the fractional parameters ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ on the system’s behavior, comparing them to integer-order derivatives. It has been demonstrated via the findings that the suggested technique is consistent with conventional numerical methods for integer-order systems while simultaneously providing an even higher level of precision. It is possible to demonstrate the efficacy and precision of this technique through simulations, which demonstrates that this method is useful for the investigation of complicated chaotic models.

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.