Subclass of m-quasiconformal harmonic functions in association with Janowski starlike functions

Abstract

Let’s take f(z)=h(z)+g(z)¯ which is an univalent sense-preserving harmonic functions in open unit disc D={z:|z|<1}. If f(z) fulfills |w(z)|=|g′(z)h′(z)|<m, where 0 ≤ m < 1, then f(z) is known m-quasiconformal harmonic function in the unit disc (Kalaj, 2010) [8]. This class is represented by SH(m).The goal of this study is to introduce certain features of the solution for non-linear partial differential equation f¯z¯=w(z)f(z) when |w(z)| < m, w(z)≺m2(b1−z)m2−b1¯z,h(z) ∈ S*(A, B). In such case S*(A, B) is known to be the class for Janowski starlike functions. We will investigate growth theorems, distortion theorems, jacobian bounds and coefficient ineqaulities, convex combination and convolution properties for this subclass.