Rotations of convex harmonic univalent mappings

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+e iθ g is convex in D. In this article, we first disprove a more flexible conjecture: “Let f=h+g‾ be a convex harmonic mapping in the disk D. Then there is a θ∈[0,2π)such that the function h+e iθ g is starlike in D”. 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+e iθ g 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.