Acta mathematica scientia,Series B ›› 2026, Vol. 46 ›› Issue (1): 39-61.doi: 10.1007/s10473-026-0103-8

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IMPROVEMENT ON HANKEL DETERMINANT BOUNDS FOR SPECIFIC HOLOMORPHIC FUNCTIONS

Huo TANG1,*, Muhammad ABBAS2, Reem K. ALHEFTHI3, Muhammad ARIF4   

  1. 1. School of Mathematics and Computer Sciences, Chifeng University, Chifeng 024000, China;
    2. Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan;
    3. Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia;
    4. Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan
  • Received:2024-08-21 Revised:2024-11-25 Online:2026-01-25 Published:2026-05-22
  • Contact: * Huo TANG, E-mail: tanghuo@cfxy.edu.cn
  • About author:Muhammad ABBAS, E-mail: muhammad abbas@awkum.edu.pk; Reem K. ALHEFTHI, E-mail: raseeri@ksu.edu.sa; Muhammad ARIF, E-mail: marifmaths@awkum.edu.pk
  • Supported by:
    The first author was partly supported by the NSFC (11561001), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT18-A14), the NSF of Inner Mongolia (2022MS01004, 2020MS01011), the Higher School Foundation of Inner Mongolia (NJZY20200), the Program for Key Laboratory Construction of Chifeng University (CFXYZD202004), the Research and Innovation Team of Complex Analysis and Nonlinear Dynamic Systems of Chifeng University (cfxykycxtd202005) and the Youth Science Foundation of Chifeng University (cfxyqn202133).

Abstract: In recent years, researchers have extensively investigated the Hankel determinant, which consists of coefficients appearing in a holomorphic function's Taylor-Maclaurin series. Hankel matrices are widely used in Markov processes, non-stationary signals, and other mathematical disciplines. The aim of the current research article is to first improve the bounds of coefficient-related problems by employing the well-known Carathéodory function. The problems that we are going to improve were obtained by Tang et al. The sharp estimates of the most difficult problem of geometric function theory known as the third-order Hankel determinant are also contributed here. Zalcman and Fekete-Szegö inequalities are also studied here for the defined family of holomorphic functions.

Key words: holomorphic function, Carathéodory function, Hankel determinant, Zalcman inequality

CLC Number: 

  • 30C45
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