Bi-Univalent Functions Associated with the Neutrosophic Touchard–Caputo Operator

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DOI: https://doi.org/10.28924/2291-8639-24-2026-239
Published: Aug 3, 2026
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Khalifa Al-Shaqsi, Jamal Salah, Ahmad Al Azab, Bi-Univalent Functions Associated with the Neutrosophic Touchard–Caputo Operator, Int. J. Anal. Appl., 24 (2026), 239.

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Khalifa Al-Shaqsi, Jamal Salah, Ahmad Al Azab

Abstract

This paper introduces a novel operator combining the neutrosophic Touchard (Bell polynomial) distribution with the modified Caputo fractional derivative operator, and applies it to study bi-univalent functions. The neutrosophic Touchard distribution extends both the classical Touchard moment series and the neutrosophic Poisson distribution through an indeterminate moment order \(n\geq 0\) and a neutrosophic parameter \(m_N\in[m_1,m_2]\). Within the new bi-univalent family \(\widetilde{\Upsilon}_{\Sigma}(\alpha,\lambda,\mu,\eta,\delta,m_N,n;\,e)\), which unifies Bazilevic and \(\mu\)-pseudo-starlike bi-univalent functions, we establish upper bounds for the coefficients \(|d_2|\) and \(|d_3|\) and derive the Fekete–Szegő inequality. Corollaries recover known results, including those of Santhiya–Thilagavathi [23] and Porwal [17]. A numerical case study with four figures illustrates how \(n\) tightens the bounds and how the neutrosophic interval quantifies model uncertainty.

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