Connections between Extensive Family of Harmonic Mappings and Pascal Distribution Series

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DOI: https://doi.org/10.28924/2291-8639-24-2026-333
Published: Oct 5, 2026
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Feras Yousef, Tariq Al-Hawary, Basem Frasin, Jamal Salah, Connections between Extensive Family of Harmonic Mappings and Pascal Distribution Series, Int. J. Anal. Appl., 24 (2026), 333.

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Feras Yousef, Tariq Al-Hawary, Basem Frasin, Jamal Salah

Abstract

Herein, we investigate a comprehensive harmonic family \(\Pi_{\eta}(k_{3},k_{2},k_{1},k_{0};s_{3},s_{2},s_{1},s_{0})\) defined by a coefficient inequality involving cubic polynomial weights. We establish several inclusion relationships between this family and various well-known subfamilies of harmonic mappings through the Pascal distribution series operator. By suitable specializations of the parameters \(k_{3},k_{2},k_{1},k_{0},s_{3},s_{2},s_{1},s_{0}\), several previously studied harmonic subfamilies and related inclusion results are recovered as special cases. The results provide a unified framework for studying inclusion relationships associated with the Pascal distribution series operator and may serve as a basis for further investigations of harmonic function families generated by other probability distribution series and related convolution operators.

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