Applications of $ q $-difference symmetric operator in harmonic univalent functions

Caihuan Zhang; Shahid Khan; Aftab Hussain; Nazar Khan; Saqib Hussain; Nasir Khan; Caihuan Zhang; Shahid Khan; Aftab Hussain; Nazar Khan; Saqib Hussain; Nasir Khan

doi:10.3934/math.2022042

AIMS Mathematics

2022, Volume 7, Issue 1: 667-680. doi: 10.3934/math.2022042

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Research article

Applications of $ q $-difference symmetric operator in harmonic univalent functions

1.
Department of Mathematics, Luoyang Normal University, Luoyang, Henan, China
2.
Department of Basic Sciences, Balochistan University of Enginearing & Technology (BUET), Khuzdar 89100, Pakistan
3.
Department of Mathematics, King Abdulaziz University, P.O. Box 80203 , Jeddah 21589 , Saudi Arabia
4.
Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan
5.
Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060 , Pakistan
6.
Department of Mathematics, FATA University, Akhorwal (Darra Adam Khel), FR Kohat 26000 , Pakistan

Received: 31 May 2021 Accepted: 08 September 2021 Published: 15 October 2021
MSC : Primary: 05A30, 30C45; Secondary: 11B65, 47B38

In this paper, for the first time, we apply symmetric $ q $ -calculus operator theory to define symmetric Salagean $ q $-differential operator. We introduce a new class $ \widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) $ of harmonic univalent functions $ f $ associated with newly defined symmetric Salagean $ q $-differential operator for complex harmonic functions. A sufficient coefficient condition for the functions $ f $ to be sense preserving and univalent and in the same class is obtained. It is proved that this coefficient condition is necessary for the functions in its subclass $ \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) } $ and obtain sharp coefficient bounds, distortion theorems and covering results. Furthermore, we also highlight some known consequence of our main results.
- univalent functions,
- harmonic functions,
- symmetric $ q $-derivative operator,
- symmetric Salagean $ q $-differential operator
Citation: Caihuan Zhang, Shahid Khan, Aftab Hussain, Nazar Khan, Saqib Hussain, Nasir Khan. Applications of $ q $-difference symmetric operator in harmonic univalent functions[J]. AIMS Mathematics, 2022, 7(1): 667-680. doi: 10.3934/math.2022042

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Abstract

In this paper, for the first time, we apply symmetric $ q $ -calculus operator theory to define symmetric Salagean $ q $-differential operator. We introduce a new class $ \widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) $ of harmonic univalent functions $ f $ associated with newly defined symmetric Salagean $ q $-differential operator for complex harmonic functions. A sufficient coefficient condition for the functions $ f $ to be sense preserving and univalent and in the same class is obtained. It is proved that this coefficient condition is necessary for the functions in its subclass $ \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) } $ and obtain sharp coefficient bounds, distortion theorems and covering results. Furthermore, we also highlight some known consequence of our main results.

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