On Two Broad Subfamilies of Liouville?Caputo-Type Fractional Derivatives Governed by Gregory Numbers

In this paper, by employing Liouville?Caputo-type fractional derivatives and subordination to the generating function of the Gregory coecients, we introduce two comprehensive subfamilies, denoted by Ez(Yu, , k, j) and Cz(Yu, ), within the family of bi-univalent functions, and demonstrate that these subfamilies are non-empty. We establish estimates for the initial Maclaurin coecients ja2j and ja3j, as well as for the Fekete?Szeg? functional associated with functions belonging to these classes. The originality of the proofs and the resulting characterizations are expected to inspire further investigation into these analytic bi-univalent function subfamilies.