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

Abstract

<abstract><p>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.</p></abstract>