On harmonic univalent mappings with nonzero pole

Abstract

In this article we consider the class AH(p) of all sense preserving harmonic functions f in the open unit disc D having a simple pole at z=p∈(0,1) with the normalizations f(0)=fz(0)−1=0. We first derive a sufficient condition for univalence of such functions. Next we consider the class SH0(p) which consists of all functions belonging to AH(p) and are univalent with the additional normalization fz¯(0)=0. We study the class SH0(p) in the geometric function theoretic viewpoint. As a byproduct of our investigation, we see that consideration of nonzero pole yields nontrivial lower bounds for the Taylor coefficients of such functions. We also prove a growth result for a non-vanishing subclass associated with SH0(p).