A class of bi-univalent functions subordinate to the $\beta$-Gregory function with an application

In this investigation, a generalized form of the Gregory function, called the $\beta$-Gregory function, is derived. This function includes several well-known analytic functions as special cases. We then consider a new symmetric class of analytic and bi-univalent functions related to the $\beta$-Gregory function. The geometric and analytic properties of this class are investigated, and inclusion relationships with other known subclasses are established under suitable sufficient conditions. Furthermore, we derive accurate estimates of the coefficients of this class and discuss the associated Fekete-Szegö inequality, showing how the results obtained contribute to generalizing and improving several previous works in bi-univalent functions.