Bi-univalent Function Subfamilies Associated with the (p,q)-derivative Operator Subordinate to Lucas-Balancing Polynomials

In the open disc $\{\zeta\in\mathbb{C}:|\zeta| <1\}$ $=\mathfrak{D}$, we present a family of bi-univalent functions $g(\zeta)=\zeta+\sum\limits_{j=2}^{\infty}d_j\zeta^j$ associated with the $(p,q)$-derivative operator and Lucas-Balancing polynomials. For members of this family, we obtain the upper bounds for $|d_2|$, $|d_3|$, and $|d_3-\xi d_2^2|$, $\xi \in\mathbb{R}$. The new implications of the main results are also discussed, along with relevant connections to earlier research.