Neutrosophic Bounds on Coefficients of Inequality for a Subclass of Holomorphic Functions Isra Al-Shbeil1,∗ , Wael Mahmoud Mohammad Salameh2,∗ , Saleem Ashhab3 , Biswajit Rath4 , Eada Ahmed Al-Zahrani5 1Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan 2Faculty of Information Technology, Abu Dhabi University, Abu Dhabi 59911, United Arab Emirates 3Department of Mathematics, Al-Albayt University, Mafraq 25113, Jordan 4Gitam Institute of Science, GITAM University, Visakhapatnam 530045, India 5 Basic and Applied Scientifc Research Center, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 31441, Dammam,Saudi Arabia Emails: i.shbeil@ju.edu.jo; wael.salameh@adu.ac.ae; ahhab@aabu.edu.jo; brath@gitam.edu; ealzahrani@iau.edu.sa Abstract This study investigates the second-order Hankel determinant in the context of certain analytic functions to find upper bounds, incorporating neutrosophic logic to handle uncertainty in coefficient estimation. The normalized ג and 0) = 0(ג conditions ′ (0) = 1 are analyzed through both classical and neutrosophic frameworks. We derive: • Sharp neutrosophic bounds for |H2,2,ϖ| when ϖ ∈ (1, 3 2 ] • Optimal bounds for |H2,3| at ϖ = 3 2 in G(ϖ) and Q(ϖ) • Neutrosophic logarithmic coefficient determinants with τ -ι-φ membership degrees The framework demonstrates robustness when coefficients exhibit simultaneous membership/non-membership characteristics. Keywords: Neutrosophic analysis; Caratheodory function; Upper bound; Hankel determinant; Holomorphic ´ function; Uncertainty quantification