Convolutions of harmonic right half-plane mappings

Abstract

Abstract We first prove that the convolution of a normalized right half-plane mapping with another subclass of normalized right half-plane mappings with the dilatation − z ( a + z ) / ( 1 + a z ) $ - z(a + z)/(1 + az)$ is CHD (convex in the horizontal direction) provided a = 1 $a = 1$ or − 1 ≤ a ≤ 0 $ - 1 \le a \le 0$ . Secondly, we give a simply method to prove the convolution of two special subclasses of harmonic univalent mappings in the right half-plane is CHD which was proved by Kumar et al. [1, Theorem 2.2]. In addition, we derive the convolution of harmonic univalent mappings involving the generalized harmonic right half-plane mappings is CHD. Finally, we present two examples of harmonic mappings to illuminate our main results.