  Convolutions or Hadamard products of analytic functions is a well explored
area of research and many nice results are available in literature. On the
other hand, very little is known in general about the convolutions of univalent
harmonic mappings. So, researchers started exploring properties of convolutions
of some specific univalent harmonic mappings and while doing so, they have
mostly used well known Cohn's rule or/and Schur-Cohn's algorithm, which
involves computations that are very cumbersome. The main objective of this
article is to present an alternative approach, which requires very less
computational efforts and allows us to prove more general results. Most of the
earlier known results follow as particular cases of the results proved herein.
