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Norm Estimates of the Pre-Schwarzian Derivatives for Functions with Conic-like Domains

Author

Listed:
  • Sidra Zafar

    (Department of Mathematics, Mirpur University of Science and Technology (MUST), Mirpur 10250, AJK, Pakistan)

  • Abbas Kareem Wanas

    (Department of Mathematics, College of Science, University of Al-Qadisiyah, Al Diwaniyah 58001, Al-Qadisiyah, Iraq)

  • Mohamed Abdalla

    (Mathematics Department, College of Science, King Khalid University, Abha 62529, Saudi Arabia)

  • Syed Zakar Hussain Bukhari

    (Department of Mathematics, Mirpur University of Science and Technology (MUST), Mirpur 10250, AJK, Pakistan)

Abstract

The pre-Schwarzianand Schwarzian derivatives of analytic functions f are defined in U , where U is the open unit disk. The pre-Schwarzian as well as Schwarzian derivatives are popular tools for studying the geometric properties of analytic mappings. These can also be used to obtain either necessary or sufficient conditions for the univalence of a function f . Because of the computational difficulty, the pre-Schwarzian norm has received more attention than the Schwarzian norm. It has applications in the theory of hypergeometric functions, conformal mappings, Teichmüller spaces, and univalent functions. In this paper, we find sharp norm estimates of the pre-Schwarzian derivatives of certain subfamilies of analytic functions involving some conic-like image domains. These results may also be extended to the families of strongly starlike, convex, as well as to functions with symmetric and conjugate symmetric points.

Suggested Citation

  • Sidra Zafar & Abbas Kareem Wanas & Mohamed Abdalla & Syed Zakar Hussain Bukhari, 2023. "Norm Estimates of the Pre-Schwarzian Derivatives for Functions with Conic-like Domains," Mathematics, MDPI, vol. 11(11), pages 1-14, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:11:p:2490-:d:1158157
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