Hyperbolically Lipschitz Continuity, Area Distortion, and Coefficient Estimates for (K, K′)-Quasiconformal Harmonic Mappings of the Unit Disk

2021 ◽  
Author(s):  
D. Zhong ◽  
W. Yuan
Keyword(s):  
Unit Disk ◽  
Lipschitz Continuity ◽  
Harmonic Mappings ◽  
Coefficient Estimates ◽  
Area Distortion

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.

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.

Coefficient Estimates for Certain Subclasses of m-Fold Symmetric Bi-univalent Functions Associated with Pseudo-Starlike Functions

2021 ◽  
pp. 209-223
Author(s):  
Timilehin G. Shaba ◽  
Amol B. Patil
Keyword(s):  
Unit Disk ◽  
Univalent Function ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates

In the present investigation, we introduce the subclasses $\varLambda_{\Sigma}^{m}(\eta,\leftthreetimes,\phi)$ and $\varLambda_{\Sigma}^{m}(\eta,\leftthreetimes,\delta)$ of \textit{m}-fold symmetric bi-univalent function class $\Sigma_m$, which are associated with the pseudo-starlike functions and defined in the open unit disk $\mathbb{U}$. Moreover, we obtain estimates on the initial coefficients $|b_{m+1}|$ and $|b_{2m+1}|$ for the functions belong to these subclasses and identified correlations with some of the earlier known classes.

scholarly journals Coefficient Estimates for a Subclass of Analytic Functions Associated with a Certain Leaf-Like Domain

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1334
Author(s):  
Bilal Khan ◽  
Hari M. Srivastava ◽  
Nazar Khan ◽  
Maslina Darus ◽  
Muhammad Tahir ◽  
...  
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Second Order ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Coefficient Estimates ◽  
Principle Of Subordination

First, by making use of the concept of basic (or q-) calculus, as well as the principle of subordination between analytic functions, generalization Rq(h) of the class R(h) of analytic functions, which are associated with the leaf-like domain in the open unit disk U, is given. Then, the coefficient estimates, the Fekete–Szegö problem, and the second-order Hankel determinant H2(1) for functions belonging to this class Rq(h) are investigated. Furthermore, similar results are examined and presented for the functions zf(z) and f−1(z). For the validity of our results, relevant connections with those in earlier works are also pointed out.

ON SOME SUBCLASSES OF HARMONIC MAPPINGS

2019 ◽  
Vol 101 (1) ◽  
pp. 130-140
Author(s):  
NIRUPAM GHOSH ◽  
VASUDEVARAO ALLU
Keyword(s):  
Unit Disk ◽  
Harmonic Mappings ◽  
Growth Theorem ◽  
Coefficient Bound

Let ${\mathcal{P}}_{{\mathcal{H}}}^{0}(M)$ denote the class of normalised harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying $\text{Re}\,(zh^{\prime \prime }(z))>-M+|zg^{\prime \prime }(z)|$, where $h^{\prime }(0)-1=0=g^{\prime }(0)$ and $M>0$. Let ${\mathcal{B}}_{{\mathcal{H}}}^{0}(M)$ denote the class of sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ satisfying $|zh^{\prime \prime }(z)|\leq M-|zg^{\prime \prime }(z)|$, where $M>0$. We discuss the coefficient bound problem, the growth theorem for functions in the class ${\mathcal{P}}_{{\mathcal{H}}}^{0}(M)$ and a two-point distortion property for functions in the class ${\mathcal{B}}_{{\mathcal{H}}}^{0}(M)$.

scholarly journals Lipschitz Continuity for the Solutions of Triharmonic Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Zhou Yu ◽  
Xiao Bing
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Dirichlet Boundary ◽  
Lipschitz Continuity ◽  
Boundary Value ◽  
Jordan Domain ◽  
Lipschitz Continuous ◽  
Equivalent Conditions

Let D be the unit disk in the complex plane C and denote T=∂D. Write Hom+T,∂Ω for the class of all sense-preserving homeomorphism of T onto the boundary of a C2 convex Jordan domain Ω. In this paper, five equivalent conditions for the solutions of triharmonic equations ∂z∂z¯3ω=ff∈CD¯ with Dirichlet boundary value conditions ωzz¯zz¯T=γ2∈CT,ωzz¯T=γ1∈CT and ωT=γ0∈Hom+T,∂Ω to be Lipschitz continuous are presented.

Coefficient Estimates and Quasi-Subordination Properties Associated with Certain Subclasses of Analytic and Bi-Univalent Functions

2015 ◽  
Vol 65 (3) ◽  
Author(s):  
S. P. Goyal ◽  
Rakesh Kumar
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Univalent Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Coefficient Estimates

AbstractIn the present paper, we obtain the estimates on initial coefficients of normalized analytic function f in the open unit disk with f and its inverse g = f

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