q-Harmonic mappings for which analytic part is q-convex functions

Abstract

In the present article we will examine the subclass of planar harmonic mappings. Let h(z) and g(z) are analytic functions in the open unit disc D = {z | |z| < 1} and having the power series represantation h(z) = z + a2z 2+. . . and g(z) = b1z+b2z+. . .. If f = h(z) + g(z) be the solution of the non-linear partial differential equation wq(z) = ( Dqg(z) Dqh(z) ) = f z̄ fz with |wq(z)| < 1, h(z) q-convex function, then this class is called q-harmonic mappings for which analytic part is q-convex functions and the class of such functions is denoted by SHC(q), where Dqh(z) = h(z)−h(qz) (1−q)z = fz, Dqg(z) = g(z)−g(qz) (1−q)z = f̄z̄, q ∈ (0, 1). Mathematics Subject Classification: 3045