On a problem of Bharanedhar and Ponnusamy involving planar harmonic mappings

In this paper, we give a negative answer to a problem presented by Bharanedhar and Ponnusamy (Rocky Mountain J. Math. 44: 753--777, 2014) concerning univalency of a class of harmonic mappings. More precisely, we show that for all values of the involved parameter, this class contains a non-univalent function. Moreover, several results on a new subclass of close-to-convex harmonic mappings, which is motivated by work of Ponnusamy and Sairam Kaliraj (Mediterr. J. Math. 12: 647--665, 2015), are obtained.