Janowski harmonic close-to-convex functions

Abstract

A harmonic mapping in the open unit disc D{double-struck} = {z||z| < 1} onto domain Ω* ⊂ ℂ is a complex valued harmonic function w = f(z) which maps D{double-struck} univalently Ω*. Each such mapping has a canonical representation f(z) = h(z) + g(z), where h(z) and g(z) are analytic in D{double-struck} and h(0) = g(0) = 0, and are called analytic part and co-analytic part of f respectively. One says that f is sense-preserving if it has positive Jacobian Jf(z) = |h'(z)|2 - |g'(z)|2 > 0 in D{double-struck}. Its second dilatation w(z) = g'(z)/h'(z) is then analytic in D{double-struck} with |w(z)| < 1. We obtain in the present work the growth and distortion theorems for the Janowski harmonic close-to-convex functions on the open unit disc D{double-struck} by applying the Shear method in the most general case of the analytic dilatation function, that is when w(z) = g'(z)/h'(z) ⇒ w(0) = b1. In that case the second dilatation is w(z) = φ(z)+b1/1+b1φ(z) , where φ(z) is Schwarz function. © 2014 Nilgün Turhan, Yasemin Kahramaner and Yaşar Polatog̃lu.