Linear Combinations of Univalent Harmonic Mappings Convex in the Direction of the Imaginary Axis

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

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Linear combinations of a class of harmonic univalent mappings

Filomat ◽

10.2298/fil1809111l ◽

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.

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On Convolution and Convex Combination of Harmonic Mappings

Journal of Mathematics ◽

10.1155/2021/6553600 ◽

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.

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A Class of Harmonic Starlike Functions defined by Multiplier Transformation

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.36 ◽

2020 ◽

pp. 455-469

Author(s):

Deepali Khurana ◽

Raj Kumar ◽

Sibel Yalcin

Keyword(s):

Convex Combination ◽

Univalent Functions ◽

Extreme Points ◽

Starlike Functions ◽

Harmonic Mappings ◽

Distortion Bounds ◽

Radius Of Convexity ◽

Univalent Harmonic Mappings ◽

Harmonic Starlike ◽

Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

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