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

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 symmetr...

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Published inAIMS mathematics Vol. 7; no. 1; pp. 667 - 680
Main Authors Zhang, Caihuan, Khan, Shahid, Hussain, Aftab, Khan, Nazar, Hussain, Saqib, Khan, Nasir
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2022
Subjects
harmonic functions
symmetric q-derivative operator
symmetric salagean q-differential operator
univalent functions
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ISSN2473-6988
2473-6988
DOI10.3934/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.
AbstractList 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.
Author Hussain, Saqib
Khan, Nasir
Zhang, Caihuan
Khan, Shahid
Khan, Nazar
Hussain, Aftab
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CorporateAuthor Department of Mathematics, Luoyang Normal University, Luoyang, Henan, China
Department of Mathematics, FATA University, Akhorwal (Darra Adam Khel), FR Kohat 26000 , Pakistan
Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan
Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060 , Pakistan
Department of Basic Sciences, Balochistan University of Enginearing & Technology (BUET), Khuzdar 89100, Pakistan
Department of Mathematics, King Abdulaziz University, P.O. Box 80203 , Jeddah 21589 , Saudi Arabia
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Snippet In this paper, for the first time, we apply symmetric $ q $ -calculus operator theory to define symmetric Salagean $ q $-differential operator. We introduce a...
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SubjectTerms harmonic functions
symmetric q-derivative operator
symmetric salagean q-differential operator
univalent functions
Title Applications of $ q $-difference symmetric operator in harmonic univalent functions
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