Sharpness results for a class of analytic functions defined via a generalized Balloon-shaped domain

Sarem H. Hadi; Adel Salim Tayyah; Alina Alb Lupaş; Sarem H. Hadi; Adel Salim Tayyah; Alina Alb Lupaş

doi:10.3934/math.20261238

AIMS Mathematics

2026, Volume 11, Issue 9: 31373-31395. doi: 10.3934/math.20261238

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Sharpness results for a class of analytic functions defined via a generalized Balloon-shaped domain

1.
Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq
2.
Department of Cybersecurity, College of Computer Science and Information Technology, University of Al-Qadisiyah, Diwaniyah, 58002, Iraq
3.
Department of Mathematics and Computer Science, University of Oradea, 1 Universitatii street, 410087 Oradea, Romania

Received: 05 April 2026 Revised: 07 September 2026 Accepted: 10 September 2026 Published: 23 September 2026
MSC : 30C45, 30C50

In geometric function theory, the estimation of initial coefficients for analytic functions is a fundamental topic of considerable interest to researchers. To gain a more profound understanding of these estimations, it becomes essential to identify the values that satisfy the state of equality (the sharpness result). This state represents one of the most difficult challenges in this field. For this reason, many studies that do not achieve the sharpness remain open for further investigations by other researchers. In this paper, we present an extreme value function that enables the acquisition of sharpness for a class of symmetric functions with real order. Furthermore, we verify the accuracy of the representation used to determine the initial coefficients. We also derive the sharp estimates by selecting appropriate values for some parameters affecting the extreme value functions. Finally, the sharp bounds of the initial coefficients for the inverse of analytic functions and Hankel determinants are derived.
- analytic functions,
- inverse analytic functions,
- Hankel determinants
Citation: Sarem H. Hadi, Adel Salim Tayyah, Alina Alb Lupaş. Sharpness results for a class of analytic functions defined via a generalized Balloon-shaped domain[J]. AIMS Mathematics, 2026, 11(9): 31373-31395. doi: 10.3934/math.20261238

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Abstract

In geometric function theory, the estimation of initial coefficients for analytic functions is a fundamental topic of considerable interest to researchers. To gain a more profound understanding of these estimations, it becomes essential to identify the values that satisfy the state of equality (the sharpness result). This state represents one of the most difficult challenges in this field. For this reason, many studies that do not achieve the sharpness remain open for further investigations by other researchers. In this paper, we present an extreme value function that enables the acquisition of sharpness for a class of symmetric functions with real order. Furthermore, we verify the accuracy of the representation used to determine the initial coefficients. We also derive the sharp estimates by selecting appropriate values for some parameters affecting the extreme value functions. Finally, the sharp bounds of the initial coefficients for the inverse of analytic functions and Hankel determinants are derived.

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