Convolutions of harmonic right half-plane mappings

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 C z/=.1 C az/ is CHD (convex in the horizontal direction) provided a D 1 or 1 Ä a Ä 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.

On convolutions of slanted half-plane mappings

“…where ς > 0 and studied in recent papers such as [9,10]. In this regard, considering the mapping L ς instead of the mapping K 0 , the conditions will be determined for which the results of Theorems B, C, and 3.1 hold.…”

“…There have been several publications involving the convolution of the harmonic functions that are convex in a particular direction such as [7][8][9][10][11]. Moreover, Dorff et al [7] proved…”

Convolutions of Harmonic Mappings Convex in the Horizontal Direction

“…In particular, Dorff [4] and Dorff et al [5] studied the convolution of harmonic univalent mappings in the right half-plane. For some recent investigations involving convolution of harmonic mappings, we refer the reader to [6][7][8][9][10][11][12][13].…”

Convolution of a harmonic mapping with $n$-starlike mappings and its partial sums

“…The result now follows by Theorem 2.15.Corollary 3.5. For a ≥ 6, let the function f a = h + ḡ be the harmonic mapping given in Example 2 20…”