scholarly journals Convolutions of Harmonic Mappings Convex in the Horizontal Direction

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Serkan Çakmak ◽  
Elif Yaşar ◽  
Sibel Yalçın
Keyword(s):  
Horizontal Direction ◽  
Harmonic Mappings ◽  
Vertical Strip

In this paper, we establish some results concerning the convolutions of harmonic mappings convex in the horizontal direction with harmonic vertical strip mappings. Furthermore, we provide examples illustrated graphically with the help of Maple to illuminate the results.

scholarly journals A proof of a conjecture on convolution of harmonic mappings and some related problems

2021 ◽  
Vol 73 (2) ◽  
pp. 283-288
Author(s):  
S. Yalçın ◽  
A. Ebadian ◽  
S. Azizi
Keyword(s):  
Half Plane ◽  
New Method ◽  
Harmonic Mappings ◽  
Vertical Strip

UDC 517.5 Recently, Kumar et al. proposed a conjecture concerning the convolution of a generalized right half-plane mapping with a vertical strip mapping. They have verified the above conjecture for and . Also, it has been proved only for . In this paper, by using of a new method, we settle this conjecture in the affirmative for all and . Moreover, we will use this method to prove some results on convolution of harmonic mappings. This new method simplifies calculations and shortens the proof of results remarkably.

scholarly journals The Properties of a New Subclass of Harmonic Univalent Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi-Hong Liu ◽  
Ying-Chun Li
Keyword(s):  
Harmonic Functions ◽  
Horizontal Direction ◽  
Harmonic Mappings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We introduced a new subclass of univalent harmonic functions defined by the shear construction in the present paper. First, we showed that the convolutions of two special subclass harmonic mappings are convex in the horizontal direction. Secondly, we proved a necessary and sufficient condition for the above subclass of harmonic mappings to be convex in the horizontal direction. We also presented some basic examples of univalent harmonic functions explaining the behavior of the image domains.

Linear combinations of a class of harmonic univalent mappings

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3111-3121
Author(s):  
Bo-Yong Long ◽  
Michael Dorff
Keyword(s):  
Linear Combination ◽  
Imaginary Axis ◽  
Sufficient Conditions ◽  
Harmonic Mapping ◽  
Harmonic Mappings ◽  
Linear Combinations ◽  
Vertical Strip ◽  
Planar Harmonic Mapping ◽  
Complex Valued ◽  
Planar Harmonic Mappings

A planar harmonic mapping is a complex-valued function f : U ? C of the form f (x + iy) = u(x,y) + iv(x,y), where u and v are both real harmonic. Such a function can be written as f = h + g?, where h and g are both analytic; the function ? = g'=h' is called the dilatation of f. We consider the linear combinations of planar harmonic mappings that are the vertical shears of the asymmetrical vertical strip mappings j(z) = 1/2isin?j log (1+zei?j/ 1+ze-i?j) with various dilatations, where ?j ? [?/2,?), j=1,2. We prove sufficient conditions for the linear combination of this class of harmonic univalent mappings to be univalent and convex in the direction of the imaginary axis.

A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM II

2017 ◽  
Vol 95 (3) ◽  
pp. 457-466 ◽  
Author(s):  
MAŁGORZATA MICHALSKA ◽  
ANDRZEJ M. MICHALSKI
Keyword(s):  
Complex Plane ◽  
Univalent Functions ◽  
Horizontal Direction ◽  
Harmonic Mappings ◽  
Connected Sets ◽  
Simply Connected ◽  
Univalence Criteria ◽  
Complex Valued ◽  
Planar Harmonic Mappings

We study properties of the simply connected sets in the complex plane, which are finite unions of domains convex in the horizontal direction. These considerations allow us to state new univalence criteria for complex-valued local homeomorphisms. In particular, we apply our results to planar harmonic mappings obtaining generalisations of the shear construction theorem due to Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25].

scholarly journals Convolutions of harmonic right half-plane mappings

2016 ◽  
Vol 14 (1) ◽  
pp. 789-800
Author(s):  
YingChun Li ◽  
ZhiHong Liu
Keyword(s):  
Half Plane ◽  
Horizontal Direction ◽  
Harmonic Mappings ◽  
The Right

AbstractWe 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)$ is CHD (convex in the horizontal direction) provided $a = 1$ or $ - 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.

scholarly journals On Convolution and Convex Combination of Harmonic Mappings

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Ahmad Sulaiman Ahmad El-Faqeer ◽  
Zhen Chuan Ng ◽  
Shamani Supramaniam
Keyword(s):  
Unit Disk ◽  
Imaginary Axis ◽  
Convex Combination ◽  
Univalent Functions ◽  
Sufficient Conditions ◽  
Horizontal Direction ◽  
Harmonic Mappings ◽  
The Family ◽  
Univalent Harmonic Mappings ◽  
Adequate Condition

In this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings are shown to be convex in the horizontal direction. Also, the family of univalent harmonic mappings of the unit disk onto a region convex in the direction of the imaginary axis is introduced. Sufficient conditions for convex combinations of harmonic mappings of this family to be univalently convex in the direction of the imaginary axis are obtained.

scholarly journals Convolutions of harmonic half-plane mappings with harmonic vertical strip mappings

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1843-1856 ◽  
Author(s):  
Zhihong Liu ◽  
Yueping Jiang ◽  
Yong Sun
Keyword(s):  
Half Plane ◽  
Horizontal Direction ◽  
Vertical Strip

In the present paper, we prove the convolutions of generalized harmonic right half-plane mappings with harmonic vertical strip mappings are univalent and convex in the horizontal direction. Moreover, some examples of harmonic univalent mappings convex in the horizontal direction are also constructed to illuminate the main results.

A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM

2016 ◽  
Vol 94 (1) ◽  
pp. 92-100 ◽  
Author(s):  
MAŁGORZATA MICHALSKA ◽  
ANDRZEJ M. MICHALSKI
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Univalent Functions ◽  
Horizontal Direction ◽  
Harmonic Mappings ◽  
Univalence Criterion ◽  
Natural Generalisation ◽  
Planar Harmonic Mappings

Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25] gave a simple and useful univalence criterion for harmonic functions, usually called the shear construction. However, the application of this theorem is limited to planar harmonic mappings that are convex in the horizontal direction. In this paper, a natural generalisation of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic function is a union of two sets that are convex in the horizontal direction.

Vertical Strip Breast Exam Very Effective

2005 ◽  
Vol 38 (22) ◽  
pp. 2
Author(s):  
PATRICE WENDLING
Keyword(s):  
Vertical Strip
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