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

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 D = {z : vertical bar z vertical bar < 1}. If f = h + <(g)over bar> is the solution of the non-linear partial differential equation w(q)(z) - D(q)g(z)/D(q)h(z) - (f) over bar(z) over bar /fz with vertical bar w(q)(z)vertical bar < 1, w(q)(z) (sic) b(1)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 S-HCq(b). Obviously that the class S-HCq(b) is the subclass of S-H. In this paper, we investigate properties of the class S-HCq(b) by using subordination techniques.