In this study, the generalized $(p,q)$-derivative operator is used to define a novel class of bi-univalent functions. For this class, we define constraints on the coefficients up to $|{\ell }_5|$. The functions are analyzed using a suitable operational method, which enables us to derive new bounds for the Fekete–Szegö functional, as well as explicit estimates for important coefficients like $|{\ell }_2|$ and $|{\ell }_3|$. In addition, we establish the upper bounds of the second and third Hankel determinants, providing insights into the geometrical and analytical properties of this class of functions.

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to $1+\mathrm{s}\mathrm{i}\mathrm{n}z$. Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a particular focus on the second- and third-order Hankel determinants. To illustrate the non-emptiness of the proposed class, we consider the function $\sqrt{1+tanhz}$, which maps the unit disk onto a bean-shaped domain. This function satisfies the required subordination condition and hence serves as an explicit member of the class. A graphical depiction of the image domain is provided to highlight its geometric characteristics. The results obtained in this work confirm that the class under study is non-trivial and possesses rich geometric structure, making it suitable for further development in the theory of geometric function classes and coefficient estimation problems.

In this work, we introduce a new subclass of bi-univalent functions using the $(p,q)$-derivative operator and the concept of subordination to generalized Laguerre polynomials ${\mathbb{L}}_t^{\varsigma }\left(k\right)$, which satisfy the differential equation $k{y}^{″}+(1+\varsigma -k)y^{\prime }+ty=0,$ with $1+\varsigma >0$, $k\in \mathbb{R}$, and $t\ge 0$. We focus on functions that blend the geometric features of starlike and convex mappings in a symmetric setting. The main goal is to estimate the initial coefficients of functions in this new class. Specifically, we obtain sharp upper bounds for $|a_2|$ and $|a_3|$ and for the Fekete–Szegö functional $|a_3-\eta a_2^2|$ for some real number $\eta$. In the final section, we explore several special cases that arise from our general results. These results contribute to the ongoing development of bi-univalent function theory in the context of $(p,q)$-calculus.

In this paper, we introduce a new subclass of analytic functions denoted byMp,qΣ (∝, β), where we use the subordination relationship between the Mittag-Leffler function and the (p, q)-derivative of F(z) to define this new class. By employing the Taylor-Maclaurin series expansion, we focus on estimating the bounds for the coefficients |a2| and |a3|. Moreover, we establish Fekete Szeg¨o inequalities for functions within this class.

In this paper, we introduce a new subclass of bi-univalent functions of complex order, using the symmetric q-derivative with subordination principles. We obtain upper bounds for the coefficients |c2|, |c3| and estimate an upper bound for the Fekete–Szegő problem within this new subclass ˜MΣq(t,ϱ,ζ).

In this paper, we introduce a novel operator formed by the convolution of the Jung-Kim-Srivastav operator and the Wright function. This operator is used to construct classes of analytic functions within the open unit disk. The primary objective is to establish precise estimates for the basic coefficients |a2| and |a3| in the Taylor-Maclaurin series for functions belonging to this class. Furthermore, we present new results related to this study.