Problems involving combinations of coefficients for the inverse of some complex-valued analytical functions

Huo Tang; Muhammad Abbas; Reem K. Alhefthi; Muhammad Arif; Huo Tang; Muhammad Abbas; Reem K. Alhefthi; Muhammad Arif

doi:10.3934/math.20241404

AIMS Mathematics

2024, Volume 9, Issue 10: 28931-28954. doi: 10.3934/math.20241404

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Problems involving combinations of coefficients for the inverse of some complex-valued analytical functions

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
^† ORCID Number: https://orcid.org/0000-0003-1484-7643

Received: 11 August 2024 Revised: 20 September 2024 Accepted: 23 September 2024 Published: 14 October 2024
MSC : 30C45, 30C50

Inequalities are essential in solving mathematical problems in many different areas of mathematics. Among these, problems involving coefficient combinations that occurred in the Taylor–Maclaurin series of the inverse of complex-valued analytic functions are the challenging ones to solve. In the current article, our aim is to study certain coefficient-related problems that construct from coefficients of the inverse of specific analytic functions. These problems include the Zalcman and Fekete–Szegö inequalities, as well as sharp estimates of the second and third-order Hankel determinants with inverse function coefficients. Also, one of the obtained results gives an improvement of the problem that has been recently published in the journal "AIMS Mathematics".
- symmetric starlike and symmetric convex functions,
- inverse functions,
- Zalcman and Fekete–Szegöinequalities,
- Hankel determinant,
- Schwarz function
Citation: Huo Tang, Muhammad Abbas, Reem K. Alhefthi, Muhammad Arif. Problems involving combinations of coefficients for the inverse of some complex-valued analytical functions[J]. AIMS Mathematics, 2024, 9(10): 28931-28954. doi: 10.3934/math.20241404

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Abstract

Inequalities are essential in solving mathematical problems in many different areas of mathematics. Among these, problems involving coefficient combinations that occurred in the Taylor–Maclaurin series of the inverse of complex-valued analytic functions are the challenging ones to solve. In the current article, our aim is to study certain coefficient-related problems that construct from coefficients of the inverse of specific analytic functions. These problems include the Zalcman and Fekete–Szegö inequalities, as well as sharp estimates of the second and third-order Hankel determinants with inverse function coefficients. Also, one of the obtained results gives an improvement of the problem that has been recently published in the journal "AIMS Mathematics".

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