scholarly journals Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains

2014 ◽  
Vol 39 ◽  
pp. 811-830 ◽  
Author(s):  
Dariusz Partyka ◽  
Ken-ichi Sakan
Keyword(s):  
Unit Disk ◽  
Harmonic Mappings ◽  
Convex Domains ◽  
Bounded Convex

scholarly journals Quasiconformal harmonic mappings and close-to-convex domains

Filomat ◽  
2010 ◽  
Vol 24 (1) ◽  
pp. 63-68 ◽  
Author(s):  
David Kalaj
Keyword(s):  
Unit Disk ◽  
Convex Domain ◽  
Harmonic Mapping ◽  
Harmonic Mappings ◽  
Convex Domains ◽  
Primary 30C55 ◽  
Subject Classifications ◽  
Mathematics Subject Classifications ◽  
Secondary 31C05

Let f = h + ? be a univalent sense preserving harmonic mapping of the unit disk U onto a convex domain ?. It is proved that: for every a such that |a| < 1 (resp. |a| = 1) the mapping ?a = h + a? is an |a| quasiconformal (a univalent) close-to-convex harmonic mapping. This gives an answer to a question posed by Chuaqui and Hern?ndez (J. Math. Anal. Appl. (2007)). 2010 Mathematics Subject Classifications. Primary 30C55, Secondary 31C05. .

On the valence of some classes of harmonic maps

1991 ◽  
Vol 110 (2) ◽  
pp. 313-325 ◽  
Author(s):  
Abdallah Lyzzaik
Keyword(s):  
Unit Disc ◽  
Harmonic Maps ◽  
Harmonic Mappings ◽  
Convex Domains ◽  
Bounded Convex

AbstractWe give examples which (i) disprove a conjecture of Sheil-Small regarding the valence of harmonic mappings of the unit disc to bounded convex domains, and (ii) answer negatively a question of the author regarding the valence of harmonic mappings with polynomial analytic and co-analytic parts.

Valency of Harmonic Mappings onto Bounded Convex Domains

2001 ◽  
Vol 1 (2) ◽  
pp. 479-499
Author(s):  
Daoud Bshouty ◽  
Walter Hengartner ◽  
Abdallah Lyzzaik ◽  
Allen Weitsman
Keyword(s):  
Harmonic Mappings ◽  
Convex Domains ◽  
Bounded Convex

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.

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 The Lindelöf principle and angular derivatives in convex domains of finite type

2002 ◽  
Vol 73 (2) ◽  
pp. 221-250 ◽  
Author(s):  
Marco Abate ◽  
Roberto Tauraso
Keyword(s):  
Finite Type ◽  
Boundary Behaviour ◽  
Convex Domains ◽  
Kobayashi Distance ◽  
Angular Derivatives ◽  
Circular Domains ◽  
Lindelöf Principle ◽  
Real Analytic ◽  
Bounded Convex ◽  
Carathéodory Theorem

AbstractWe describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelöf principle.

Sign in / Sign up

Export Citation Format

Share Document