Certain new applications of Faber polynomial expansion for some new subclasses of $ \upsilon $-fold symmetric bi-univalent functions associated with $ q $-calculus

Mohammad Faisal Khan; Mohammad Faisal Khan

doi:10.3934/math.2023521

AIMS Mathematics

2023, Volume 8, Issue 5: 10283-10302. doi: 10.3934/math.2023521

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Research article

Certain new applications of Faber polynomial expansion for some new subclasses of $ \upsilon $-fold symmetric bi-univalent functions associated with $ q $-calculus

Mohammad Faisal Khan ^,

Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia

Received: 09 January 2023 Revised: 06 February 2023 Accepted: 13 February 2023 Published: 28 February 2023
MSC : Primary 05A30, 30C45; Secondary 11B65, 47B38

In this article, we define the $ q $-difference operator and Salagean $ q $-differential operator for $ \upsilon $-fold symmetric functions in open unit disk $ \mathcal{U} $ by first applying the concepts of $ q $-calculus operator theory. Then, we considered these operators in order to construct new subclasses for $ \upsilon $-fold symmetric bi-univalent functions. We establish the general coefficient bounds $ |a_{\upsilon k+1}| $ for the functions in each of these newly specified subclasses using the Faber polynomial expansion method. Investigations are also performed on Feketo-Sezego problems and initial coefficient bounds for the function $ h $ that belong to the newly discovered subclasses. To illustrate the relationship between the new and existing research, certain well-known corollaries of our main findings are also highlighted.
Citation: Mohammad Faisal Khan. Certain new applications of Faber polynomial expansion for some new subclasses of $ \upsilon $-fold symmetric bi-univalent functions associated with $ q $-calculus[J]. AIMS Mathematics, 2023, 8(5): 10283-10302. doi: 10.3934/math.2023521

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Abstract

In this article, we define the $ q $-difference operator and Salagean $ q $-differential operator for $ \upsilon $-fold symmetric functions in open unit disk $ \mathcal{U} $ by first applying the concepts of $ q $-calculus operator theory. Then, we considered these operators in order to construct new subclasses for $ \upsilon $-fold symmetric bi-univalent functions. We establish the general coefficient bounds $ |a_{\upsilon k+1}| $ for the functions in each of these newly specified subclasses using the Faber polynomial expansion method. Investigations are also performed on Feketo-Sezego problems and initial coefficient bounds for the function $ h $ that belong to the newly discovered subclasses. To illustrate the relationship between the new and existing research, certain well-known corollaries of our main findings are also highlighted.

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