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 ${\tilde{\mathcal{H}}}_q^m\left(\alpha \right)$ 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 $\overline{{\tilde{\mathcal{H}}}_q^m\left(\alpha \right)}$ and obtain sharp coefficient bounds, distortion theorems and covering results. Furthermore, we also highlight some known consequence of our main results.