Blaschke Products and Convolutions with a Slanted Generalized Half-Plane Harmonic Mapping

Abstract

It is known that if the convolution of two suitably normalized planar harmonic mappings from certain families of mappings, such as those mapping into a half-plane or a strip, is locally univalent, the convolution is univalent and convex in one direction. After extending this to convolutions of mappings into a slanted half-plane with those into a slanted asymmetric strip, we prove properties for the dilatation of the convolution of a mapping from a family of slanted generalized right half-plane mappings with mappings into a slanted half-plane or a slanted asymmetric strip with a finite Blaschke product dilatation. The properties lay the foundation for a direct application of polynomial zero distribution techniques in the determination of local univalence of such convolutions. We conclude by producing a family of univalent convolutions convex in one direction between a slanted generalized right half-plane mapping and a mapping into a half-plane with a two-factor Blaschke product dilatation.