$q-$Harmonic mappings for which analytic part is $q-$convex functions of complex order

We introduce a new class of harmonic function $f$, that is subclass of planar harmonic mapping associated with $q-$difference operator. Let $h$ and $g$ are analytic functions in the open unit disc $\mathbb{D}=\{ z\,:\,|z|<1 \}$. If $f=h+\bar{g}$ is the solution of the non-linear partial differential equation $w_q(z)=\dfrac{D_q g(z)}{D_q h(z)}=\dfrac{\bar{f}_\bar{z}}{f_z}$ with $|w_q(z)|<1$, $w_q(z)\prec b_1 \dfrac{1+z}{1-qz}$ and $h$ is $q-$convex function of complex order, then the class of such functions are called $q-$harmonic functions for which analytic part is $q-$convex functions of complex order denoted by $\mathcal{S}_{ \mathcal{H}\mathcal{C}_q(b)}$. Obviously that the class $\mathcal{S}_{ \mathcal{H}\mathcal{C}_q(b)}$ is the subclass of $\mathcal{S}_\mathcal{H}$. In this paper, we investigate properties of the class $\mathcal{S}_{ \mathcal{H}\mathcal{C}_q(b)}$ by using subordination techniques.

Keywords:

$q-$difference operator $q-$harmonic mapping, $q-$convex function of complex order,

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