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 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.

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.

scholarly journals Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nam Q. Le
Keyword(s):  
Boundary Data ◽  
Boundary Regularity ◽  
Hölder Regularity ◽  
Convex Domains ◽  
Open Convex ◽  
Optimal Boundary ◽  
Monge Ampere ◽  
Holder Regularity ◽  
Bounded Convex

<p style='text-indent:20px;'>By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as <inline-formula><tex-math id="M1">\begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document}</tex-math></inline-formula> with zero boundary data, have unexpected degenerate nature.</p>

scholarly journals Harmonic mappings onto R-convex domains

2019 ◽  
Vol 26 (2) ◽  
pp. 37-50
Author(s):  
S. Yu. Graf
Keyword(s):  
Harmonic Mappings ◽  
Convex Domains

Harmonic Mappings onto Convex Domains

1987 ◽  
Vol 39 (6) ◽  
pp. 1489-1530 ◽  
Author(s):  
Yusuf Abu-Muhanna ◽  
Glenn Schober
Keyword(s):  
Univalent Functions ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Convex Domains ◽  
Harmonic Solution ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected ◽  
Complex Valued

Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equationwhere a is analytic and |a| < 1, such that f(U) ⊂ D and

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