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

• Published in: AIMS 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
• Language: English
• Published: AIMS Press 01.01.2022
• Subjects: harmonic functions; symmetric q-derivative operator; symmetric salagean q-differential operator; univalent functions
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2022042

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.