On convolutions of slanted half-plane mappings

Abstract

The convolution of convex harmonic univalent functions in the unit disk, unlike analytic functions, may not be convex or even univalent. The main purpose of this work is to develop previous work involving the convolution of convex harmonic functions. Briefly, we obtain under which conditions the convolution of a right half-plane harmonic mapping having a dilatation -z and a slanted half-plane harmonic mapping with beta having a dilatation e(i mu)rho+z/1+rho z (|rho| < 1 and mu is an element of R) is univalent and convex in the direction -beta. We also provide an example illustrating graphically with the help of Maple to illuminate the result.