# Convolutions of harmonic right half-plane mappings

Open Mathematics (2016)

• Volume: 14, Issue: 1, page 789-800
• ISSN: 2391-5455

top

## Abstract

top
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+az) $-z\left(a+z\right)/\left(1+az\right)$ 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.

## How to cite

top

YingChun Li, and ZhiHong Liu. "Convolutions of harmonic right half-plane mappings." Open Mathematics 14.1 (2016): 789-800. <http://eudml.org/doc/287155>.

@article{YingChunLi2016,
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+az) $- z(a + z)/(1 + az)$ 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.},
author = {YingChun Li, ZhiHong Liu},
journal = {Open Mathematics},
keywords = {Harmonic univalent mappings; Harmonic convolution; Generalized right half-plane mappings; harmonic mappings; convolution},
language = {eng},
number = {1},
pages = {789-800},
title = {Convolutions of harmonic right half-plane mappings},
url = {http://eudml.org/doc/287155},
volume = {14},
year = {2016},
}

TY - JOUR
AU - YingChun Li
AU - ZhiHong Liu
TI - Convolutions of harmonic right half-plane mappings
JO - Open Mathematics
PY - 2016
VL - 14
IS - 1
SP - 789
EP - 800
AB - 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+az) $- z(a + z)/(1 + az)$ 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.
LA - eng
KW - Harmonic univalent mappings; Harmonic convolution; Generalized right half-plane mappings; harmonic mappings; convolution
UR - http://eudml.org/doc/287155
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.