On the Shears of Univalent Harmonic Mappings

Abstract

Let S be the standard class of normalized, univalent, analytic functions of the open unit disc D, and let SH0 be the class of sense-preserving, univalent, harmonic mappings f= h+ g¯ of D, where h(z)=z+∑n=2∞anznandg(z)=∑n=2∞bnzn.The purpose of this article is to disprove a conjecture by S. Ponnusamy and A. Sairam Kaliraj asserting that for every function f=h+g¯∈SH0, there exists a value θ∈ R such that the function h+ ei θg∈ S. © 2018, Springer Nature Switzerland AG.