$q$-analogue of a $p$-harmonic mapping

The primary objective of this paper is to explore a novel subclass ${{\mathscr{H}}{𝒮}}_p(q,\alpha )$ of $p$-harmonic mappings, along with the associated subclass ${{\mathscr{H}}{𝒮}}_p^0(q,\alpha )$. We demonstrate that the mapping ${{\mathscr{H}}{𝒮}}_p(q,\alpha )$ is both univalent and sense-preserving within the unit disk $𝕌$. Furthermore, we determine the extreme points of ${{\mathscr{H}}{𝒮}}_p^0(q,\alpha )$ and establish that ${{\mathscr{H}}{𝒮}}_p(q,\alpha )\cap {{𝒯}}_p$ is defined. Additional results include the derivation of distortion bounds, the convolution condition, and the convex combination for this subclass. Finally, we examine the class-preserving integral operator and introduce a $q$-Jackson type integral operator.