Third-Order Hankel Determinant for a Class of Bi-Univalent Functions Associated with Sine Function

Mohammad El-Ityan, Mustafa A. Sabri, Suha Hammad, Basem Frasin, Tariq Al-Hawary () and Feras Yousef
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Mohammad El-Ityan: Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan
Mustafa A. Sabri: Department of Mathematics, College of Education, Mustansiriyah University, Baghdad 10052, Iraq
Suha Hammad: Department of Mathematics, College of Education for Pure Sciences, University of Tikrit, Tikrit 34001, Iraq
Basem Frasin: Department of Mathematics, Faculty of Science, Al Al-Bayt University, Mafraq 25113, Jordan
Tariq Al-Hawary: Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan
Feras Yousef: Department of Mathematics, The University of Jordan, Amman 11942, Jordan

Mathematics, 2025, vol. 13, issue 17, 1-15

Abstract: This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to 1 + s i n z . Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a particular focus on the second- and third-order Hankel determinants. To illustrate the non-emptiness of the proposed class, we consider the function 1 + tanh z , which maps the unit disk onto a bean-shaped domain. This function satisfies the required subordination condition and hence serves as an explicit member of the class. A graphical depiction of the image domain is provided to highlight its geometric characteristics. The results obtained in this work confirm that the class under study is non-trivial and possesses rich geometric structure, making it suitable for further development in the theory of geometric function classes and coefficient estimation problems.

Keywords: bean-shaped; Fekete-Szegö; starlike functions; Hankel determinants; bi-univalent functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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