Rotations of convex harmonic univalent mappings

Abstract

Let f=h+g‾ be a normalized and sense-preserving convex harmonic mapping in the unit disk D. In a recent paper, Ponnusamy and Sairam Kaliraj conjectured that there is a θ∈[0,2π) such that the function h+eiθg is convex in D. In this article, we first disprove a more flexible conjecture: “Letf=h+g‾be a convex harmonic mapping in the diskD. Then there is aθ∈[0,2π)such that the functionh+eiθgis starlike inD”. In addition, we present an example to show that there exists a harmonic automorphism f=h+g‾ of a disk such that the function h+eiθg is convex in only one direction for θ≠0, and that the analytic function h+g is not starlike therein. The article concludes with a new conjecture.