Study of quantum calculus for a new subclass of $ q $-starlike bi-univalent functions connected with vertical strip domain

Ahmad A. Abubaker; Khaled Matarneh; Mohammad Faisal Khan; Suha B. Al-Shaikh; Mustafa Kamal; Ahmad A. Abubaker; Khaled Matarneh; Mohammad Faisal Khan; Suha B. Al-Shaikh; Mustafa Kamal

doi:10.3934/math.2024577

AIMS Mathematics

2024, Volume 9, Issue 5: 11789-11804. doi: 10.3934/math.2024577

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Research article

Study of quantum calculus for a new subclass of $ q $-starlike bi-univalent functions connected with vertical strip domain

1.
Faculty of Computer Studies, Arab Open University, Saudi Arabia
2.
Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh-11673, Saudi Arabia

Received: 26 December 2023 Revised: 26 February 2024 Accepted: 13 March 2024 Published: 26 March 2024
MSC : Primary 30A55, 30C45, Secondary 11B65, 47B38

In this study, using the ideas of subordination and the quantum-difference operator, we established a new subclass $ \mathcal{S} ^{\ast }\left(\delta, \sigma, q\right) $ of $ q $-starlike functions and the subclass $ \mathcal{S}_{\Sigma }^{\ast }\left(\delta, \sigma, q\right) $ of $ q $-starlike bi-univalent functions associated with the vertical strip domain. We examined sharp bounds for the first two Taylor-Maclaurin coefficients, sharp Fekete-Szegö type problems, and coefficient inequalities for the function $ h $ that belong to $ \mathcal{S}^{\ast }\left(\delta, \sigma, q\right) $, as well as sharp bounds for the inverse function $ h $ that belong to $ \mathcal{S}^{\ast }\left(\delta, \sigma, q\right) $. We also investigated some results for the class of bi-univalent functions $ \mathcal{S}_{\Sigma }^{\ast }\left(\delta, \sigma, q\right) $ and well-known corollaries were also highlighted to show connections between previous results and the findings of this paper.
- analytic functions,
- bi-univalent functions,
- quantum-calculus,
- vertical strip domain,
- $ q $-starlike functions,
- subordination,
- $ q $-difference operator
Citation: Ahmad A. Abubaker, Khaled Matarneh, Mohammad Faisal Khan, Suha B. Al-Shaikh, Mustafa Kamal. Study of quantum calculus for a new subclass of $ q $-starlike bi-univalent functions connected with vertical strip domain[J]. AIMS Mathematics, 2024, 9(5): 11789-11804. doi: 10.3934/math.2024577

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Abstract

In this study, using the ideas of subordination and the quantum-difference operator, we established a new subclass $ \mathcal{S} ^{\ast }\left(\delta, \sigma, q\right) $ of $ q $-starlike functions and the subclass $ \mathcal{S}_{\Sigma }^{\ast }\left(\delta, \sigma, q\right) $ of $ q $-starlike bi-univalent functions associated with the vertical strip domain. We examined sharp bounds for the first two Taylor-Maclaurin coefficients, sharp Fekete-Szegö type problems, and coefficient inequalities for the function $ h $ that belong to $ \mathcal{S}^{\ast }\left(\delta, \sigma, q\right) $, as well as sharp bounds for the inverse function $ h $ that belong to $ \mathcal{S}^{\ast }\left(\delta, \sigma, q\right) $. We also investigated some results for the class of bi-univalent functions $ \mathcal{S}_{\Sigma }^{\ast }\left(\delta, \sigma, q\right) $ and well-known corollaries were also highlighted to show connections between previous results and the findings of this paper.

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