Convolutions of Planar Harmonic Convex Mappings

Ruscheweyh and Sheil-Small proved that convexity is preserved under the convolution of univalent analytic mappings in K. However, when we consider the convolution of univalent harmonic convex mappings in K H , this property does not hold. In fact, such convolutions may not be univalent. We establish some results concerning the convolution of univalent harmonic convex mappings provided that it is locally univalent. In particular, we show that the convolution of a right half-plane mapping in K H with either another right halfplane mapping or a vertical strip mapping in K H is convex in the direction of the real axis. Further, we give a condition under which the convolution of a vertical strip mapping in K H with itself will be convex in the direction of the real axis.