scholarly journals Some properties of univalent log-harmonic mappings

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5275-5288 ◽  
Author(s):  
Zhihong Liu ◽  
Saminathan Ponnusamy
Keyword(s):  
Representation Theorem ◽  
Distortion Theorem ◽  
Harmonic Mappings ◽  
Schwarzian Derivatives ◽  
Starlike Mappings ◽  
Harmonic Starlike ◽  
Coefficients Estimate

We determine the representation theorem, distortion theorem, coefficients estimate and Bohr?s radius for log-harmonic starlike mappings of order ?, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of 1-slit log-harmonic mappings. Finally, we introduce pre-Schwarzian, Schwarzian derivatives and Bloch?s norm for non-vanishing log-harmonic mappings, several properties related to these are also obtained.

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

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 $\overline{HS}(k,\lambda,b,\alpha)$.

scholarly journals Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

2013 ◽  
Vol 25 (1) ◽  
pp. 64-91 ◽  
Author(s):  
Rodrigo Hernández ◽  
María J. Martín
Keyword(s):  
Harmonic Mappings ◽  
Schwarzian Derivatives ◽  
Derivatives Of

Distortion Theorems for Diffeomorphisms

1994 ◽  
Vol 37 (3) ◽  
pp. 351-360
Author(s):  
W. Hengartner ◽  
D. Poulin
Keyword(s):  
Distortion Theorem ◽  
Harmonic Mappings ◽  
Univalent Harmonic Mappings ◽  
Distortion Theorems

AbstractGeneralizations of the Koebe distortion theorem to a class of diffeomorphisms are given. They are applied to univalent harmonic mappings.

scholarly journals On Harmonic Functions Defined by Differential Operator with Respect tok-Symmetric Points

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Afaf A. Ali Abubaker ◽  
Maslina Darus
Keyword(s):  
Differential Operator ◽  
Representation Theorem ◽  
Harmonic Functions ◽  
Univalent Functions ◽  
Distortion Theorem ◽  
Coefficient Condition ◽  
Symmetric Points

We introduce new classesMHkσ,s(λ,δ,α)andM¯Hkσ,s(λ,δ,α)of harmonic univalent functions with respect tok-symmetric points defined by differential operator. We determine a sufficient coefficient condition, representation theorem, and distortion theorem.

scholarly journals Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for -quasiconformal harmonic mappings of unit disk

2021 ◽  
Vol 73 (2) ◽  
pp. 151-159
Author(s):  
Deguang Zhong ◽  
Wenjun Yuan
Keyword(s):  
Unit Disk ◽  
Lipschitz Continuity ◽  
Distortion Theorem ◽  
Harmonic Mappings ◽  
Coefficient Estimate ◽  
Coefficient Estimates ◽  
Area Distortion

UDC 517.51 We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem,  and coefficient estimate for the classes of -quasiconformal harmonic mappings from the unit disk onto itself.

scholarly journals Coefficients estimate for a subclass of holomorphic mappings on the unit polydisk in Cn

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 27-36
Author(s):  
Liangpeng Xiong
Keyword(s):  
Holomorphic Mappings ◽  
High Dimensional ◽  
Dimensional Version ◽  
Unit Polydisk ◽  
Starlike Mappings ◽  
Coefficients Estimate

The aim of this paper is to obtain the sharp solutions of Fekete-Szeg? problems of high dimensional version for family of holomorphic mappings that are normalized on the unit polydisk Un in Cn. The main results unify some recent works, which are closely related to the starlike mappings. Moreover, some previous results are improved.

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

2020 ◽  
pp. 445-454
Author(s):  
Deepali Khurana ◽  
Sushma Gupta ◽  
Sukhjit Singh
Keyword(s):  
Present Article ◽  
Unit Disk ◽  
Imaginary Axis ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Distortion Bounds ◽  
Univalent Harmonic Mappings

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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