On the Shears of Univalent Harmonic Mappings

Abstract

Let S be the standard class of normalized, univalent, analytic functions of the open unit disc $$\mathbb {D},$$ and let $$S_H^0$$ be the class of sense-preserving, univalent, harmonic mappings $$f=h + \overline{g}$$ of $$\mathbb {D},$$ where $$\begin{aligned} h(z) = z+\sum _{n=2}^{\infty } a_nz^n\;\;\;\; \mathrm{and}\;\;\;\; g(z) = \sum _{n=2}^{\infty } b_nz^n. \end{aligned}$$ The purpose of this article is to disprove a conjecture by S. Ponnusamy and A. Sairam Kaliraj asserting that for every function $$f=h + \overline{g}\in S_H^0,$$ there exists a value $$\theta \in \mathbb {R}$$ such that the function $$h+ e^{i\theta }g\in S.$$