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

Authors : Asena ÇETİNKAYA, Yaşar POLATOĞLU

Pages : 813-820

View : 18 | Download : 5

Publication Date : 2018-08-01

Article Type : Research Paper

Abstract :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 $w_qinsert ignore into journalissuearticles values(z);=\dfrac{D_q ginsert ignore into journalissuearticles values(z);}{D_q hinsert ignore into journalissuearticles values(z);}=\dfrac{\bar{f}_\bar{z}}{f_z}$ with $|w_qinsert ignore into journalissuearticles values(z);|<1$, $w_qinsert ignore into journalissuearticles values(z);\prec b_1 \dfrac{1+z}{1-qz}$ 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 $\mathcal{S}_{ \mathcal{H}\mathcal{C}_qinsert ignore into journalissuearticles values(b);}$. Obviously that the class  $\mathcal{S}_{ \mathcal{H}\mathcal{C}_qinsert ignore into journalissuearticles values(b);}$ is the subclass of $\mathcal{S}_\mathcal{H}$. In this paper, we investigate properties of the class  $\mathcal{S}_{ \mathcal{H}\mathcal{C}_qinsert ignore into journalissuearticles values(b);}$ by using subordination techniques.
Keywords : q difference operator, q harmonic mapping, q convex function of complex order