The results presented in this paper deal with the classical but still prevalent problem of introducing new classes of m-fold symmetric bi-univalent functions and studying properties related to coefficient estimates. Quantum calculus aspects are also considered in this study in order to enhance its novelty and to obtain more interesting results. We present three new classes of bi-univalent functions, generalizing certain previously studied classes. The relation between the known results and the new ones presented here is highlighted. Estimates on the Taylor–Maclaurin coefficients $|a_{m+1}|$ and $|a_{2m+1}|$ are obtained and, furthermore, the much investigated aspect of Fekete–Szegő functional is also considered for each of the new classes.

Using the (p, q)-derivative operator we introduce new subclasses of analytic and bi-univalent functions, we obtain estimates on coefficients and the Fekete-Szegö functional.

The motivation of the present article is to define the $(p-q)$-Wanas operator in geometric function theory by the symmetric nature of quantum calculus. We also initiate and explore certain new families of holormorphic and bi-univalent functions ${\mathcal{A}}_E(\lambda ,\sigma ,\delta ,s,t,p,q;\vartheta )$ and $S_E(\mu ,\gamma ,\sigma ,\delta ,s,t,p,q;\vartheta )$ which are defined in the unit disk U associated with the $(p-q)$-Wanas operator. The upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö-type inequalities for the functions in these families are obtained. Furthermore, several consequences of our results are pointed out based on the various special choices of the involved parameters.

This paper investigates the third Hankel determinant, denoted $H_3\left(1\right)$, for functions within the subclass $\mathcal{R}{\mathcal{S}}_{\sum }^{\mathrm{*}}\left(\lambda \right)$ of bi-univalent functions associated with crescent-shaped regions ${\phi }_{⦅}\left(z\right)=z+\sqrt{1+z^2}$. The primary aim of this study is to establish upper bounds for $H_3\left(1\right)$. By analyzing functions within this specific geometric context, we derive precise constraints on the determinant, thereby enhancing our understanding of its behavior. Our results and examples provide valuable insights into the properties of bi-univalent functions in crescent-shaped domains and contribute to the broader theory of analytic functions.

In the present paper we introduce a new class of analytic functions f in the open unit disk normalized by f (0) = f ^′ (0) − 1 = 0, associated with exponential functions. The aim of the present paper is to investigate the third-order Hankel determinant H ₃ (1) for this function class and obtain the upper bound of the determinant H ₃ (1).

In the current article, making use of certain operator, we initiate and explore a certain family ${\mathcal{W}}_{\Sigma }(\lambda ,\gamma ,\sigma ,\delta ,\alpha ,\beta ,p,q;h)$ of holomorphic and bi-univalent functions in the open unit disk $\mathbb{D}$. We establish upper bounds for the initial Taylor–Maclaurin coefficients and the Fekete–Szegö type inequality for functions in this family.