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

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DOI: https://doi.org/10.28924/2291-8639-24-2026-87
Published: Mar 19, 2026
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Feras Yousef, Tariq Al-Hawary, Jamal Salah, Basem Aref Frasin, Mohamed Illafe, On Two Broad Subfamilies of Liouville-Caputo-Type Fractional Derivatives Governed by Gregory Numbers, Int. J. Anal. Appl., 24 (2026), 87.

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Feras Yousef, Tariq Al-Hawary, Jamal Salah, Basem Aref Frasin, Mohamed Illafe

Abstract

In this paper, by employing Liouville--Caputo-type fractional derivatives and subordination to the generating function of the Gregory coefficients, we introduce two comprehensive subfamilies, denoted by \(E_{\digamma}(\Psi_{u},\rho,\) \(\daleth,\) \(\gimel)\) and \(C_{\digamma}(\Psi_{u},\phi)\), within the family of bi-univalent functions, and demonstrate that these subfamilies are non-empty. We establish estimates for the initial Maclaurin coefficients \(\left\vert a_{2}\right\vert \) and \(\left\vert a_{3}\right\vert \), as well as for the Fekete--Szegö 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.

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