On the Shears of Univalent Harmonic Mappings

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 \(h+ e^{i\theta }g\in S.\)