Title: On a problem of Bharanedhar and Ponnusamy involving planar harmonic mappings,
Abstract: 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.