CONSTRUCTION OF UNIVALENT HARMONIC MAPPINGS AND THEIR CONVOLUTIONS

Abstract

In this article, we make use of  convex analytic functions $H_a(z)=[1/(1-a)]\log[(1-az)/(1-z)]$, $a\in \mathbb{R}$, $|a|\leq 1$, $a\neq 1$ and starlike analytic functions $L_b(z)=z/[(1-bz)(1-z)]$, $b\in \mathbb{R}$, $|b|\leq 1$, to construct  univalent harmonic functions by means of a transformation on some normalized univalent analytic functions. Besides exploring mapping properties of harmonic functions so constructed, we establish sufficient conditions for their harmonic  convolutions or Hadamard products to be locally univalent and sense preserving, univalent and convex in some direction.