New comprehensive two subclasses related to Gregory numbers of analytic bi-univalent functions

‎In this paper‎, ‎using subordinations with the functions whose coefficients are Gregory‎ ‎numbers‎, ‎we present two novel subclasses \(\mathbf{\wp}_{\Pi}(\vartheta\)‎, ‎\(\gamma,\) \(\beta)\)‎, ‎and \(\mathbf{‎ ‎\mathcal{F}‎ ‎}_{\Pi}(\phi)\) within the bi-univalent function family‎. ‎We study the estimates‎ ‎\(\left\vert a_{2}\right\vert \) and \(\left\vert a_{3}\right\vert \) of the‎ ‎Maclaurin coefficients and the Fekete-Szego inequality regarding‎ ‎functions in every one of these two subclasses‎. ‎Following the originality of‎ ‎the characterizations and the proofs may encourage additional research on‎ ‎these kinds of similarly defined analytic bi-univalent function subclasses‎.