Application of Three Leaf Domain on a Subclass of Bi-univalent Functions

In this manuscript, our inspiration stems from recent advancements in research and the widely recognized notion of coefficient estimates applicable to analytic and bi-univalent functions(BF) categories. Initially, we introduce fresh subcategories, denoted as $\mathcal{T}{\mathcal{D}_\Sigma }$, within the realm of analytic functions(AF) and BF. These subcategories are intricately linked with the concept of a three-leaf domain. Subsequently, we tackle the Fekete-Szegö problem within the scope of the $\mathcal{T}{\mathcal{D}_\Sigma }$ class that pertains to a three-leaf domain. We concentrate on setting boundaries for the coefficients, as well as defining a maximum limit for the second Hankel determinant(HD). Notably, it should be emphasized that nearly all outcomes attain their utmost precision, accompanied by the presentation of their respective extreme functions.