Investigation of the Hankel Determinant Sharp Bounds for a Specific Analytic Function Linked to a Cardioid-Shaped Domain

Isra Al-Shbeil (), Muhammad Imran Faisal, Muhammad Arif, Muhammad Abbas and Reem K. Alhefthi
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Isra Al-Shbeil: Department of Mathematics, Faculty of Science, The University of Jordon, Amman 11942, Jordan
Muhammad Imran Faisal: Department of Mathematics, Taibah University, Universities Road, P.O. Box 344, Medina 42317, Saudi Arabia
Muhammad Arif: Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
Muhammad Abbas: Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
Reem K. Alhefthi: Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Mathematics, 2023, vol. 11, issue 17, 1-22

Abstract: One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these coefficient-related problems, the problem of the third-order Hankel determinant sharp bound is the most difficult one. The aim of the present study is to determine the sharp bound of the Hankel determinant of third order by using the methodology of the aforementioned Carathéodory function family. Further, we also study some other coefficient-related problems, such as the Fekete–Szegő inequality and the second-order Hankel determinant. We examine these results for the family of bounded turning functions linked with a cardioid-shaped domain.

Keywords: Hankel determinant; coefficient bounds; bounded turning functions; cardioid domain (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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