Applications of a certain $q$-integral operator to the subclasses of analytic and bi-univalent functions

Bilal Khan; H. M. Srivastava; Muhammad Tahir; Maslina Darus; Qazi Zahoor Ahmad; Nazar Khan; Bilal Khan; H. M. Srivastava; Muhammad Tahir; Maslina Darus; Qazi Zahoor Ahmad; Nazar Khan

doi:10.3934/math.2021061

AIMS Mathematics

2021, Volume 6, Issue 1: 1024-1039. doi: 10.3934/math.2021061

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Research article

Applications of a certain $q$-integral operator to the subclasses of analytic and bi-univalent functions

1.
School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai 200241, People's Republic of China
2.
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China
4.
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan
5.
Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan
6.
Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia
7.
Government Akhtar Nawaz Khan (Shaheed) Degree College KTS, Haripur 22620, Pakistan

Received: 11 May 2020 Accepted: 14 October 2020 Published: 06 November 2020
MSC : Primary 05A30, 30C45; Secondary 11B65, 47B38

In the present investigation, our aim is to define a generalized subclass of analytic and bi-univalent functions associated with a certain $q$-integral operator in the open unit disk $\mathbb{U}$. We estimate bounds on the initial Taylor-Maclaurin coefficients $\left \vert a_{2}\right \vert$ and $\left \vert a_{3}\right \vert $ for normalized analytic functions $f$ in the open unit disk by considering the function $f$ and its inverse $g = f^{{-}{1}}$. Furthermore, we derive special consequences of the results presented here, which would apply to several (known or new) subclasses of analytic and bi-univalent functions.
Citation: Bilal Khan, H. M. Srivastava, Muhammad Tahir, Maslina Darus, Qazi Zahoor Ahmad, Nazar Khan. Applications of a certain $q$-integral operator to the subclasses of analytic and bi-univalent functions[J]. AIMS Mathematics, 2021, 6(1): 1024-1039. doi: 10.3934/math.2021061

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Abstract

In the present investigation, our aim is to define a generalized subclass of analytic and bi-univalent functions associated with a certain $q$-integral operator in the open unit disk $\mathbb{U}$. We estimate bounds on the initial Taylor-Maclaurin coefficients $\left \vert a_{2}\right \vert$ and $\left \vert a_{3}\right \vert $ for normalized analytic functions $f$ in the open unit disk by considering the function $f$ and its inverse $g = f^{{-}{1}}$. Furthermore, we derive special consequences of the results presented here, which would apply to several (known or new) subclasses of analytic and bi-univalent functions.

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