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Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions

Author

Listed:
  • Tariq Al-Hawary

    (Department of Applied Science, Ajloun College, Al Balqa Applied University, Ajloun 26816, Jordan
    These authors contributed equally to this work.)

  • Basem Frasin

    (Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq 25113, Jordan
    These authors contributed equally to this work.)

  • Ibtisam Aldawish

    (Mathematics and Statistics Department, College of Science, IMSIU (Imam Mohammad Ibn Saud Islamic University), Riyadh 13327, Saudi Arabia
    These authors contributed equally to this work.)

Abstract

In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series E ϕ ∗ ( ς , z ) to belong to the inclusive subclass Π η ( Q 3 , Q 2 , Q 1 , Q 0 ) . Necessary and sufficient conditions are also given for the generalized distribution series E ϕ ∗ ( ς , z ) ℏ and the integral operator J ς ϕ ( z ) to be in the inclusive subclass Π η ( Q 3 , Q 2 , Q 1 , 0 ) . Further, we provide a number of corollaries, which improve the existing ones that are available in some recent studies. The results presented here not only improve the earlier studies, but also give rise to a number of new results for particular choices of Q 3 , Q 2 , Q 1 and Q 0 .

Suggested Citation

  • Tariq Al-Hawary & Basem Frasin & Ibtisam Aldawish, 2025. "Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions," Mathematics, MDPI, vol. 13(12), pages 1-9, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:12:p:2029-:d:1682911
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