# Uniqueness of harmonic mappings with Blaschke dilatations

@article{Bshouty2007UniquenessOH, title={Uniqueness of harmonic mappings with Blaschke dilatations}, author={Daoud Bshouty and Abdallah Lyzzaik and Allen Weitsman}, journal={The Journal of Geometric Analysis}, year={2007}, volume={17}, pages={41-47} }

AbstractLet Ω be a bounded convex domain and let ω be a finite Blaschke product of order N = 1, 2, .... It is known that the elliptic differential equation
% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca…

## 6 Citations

Uniqueness of Harmonic Mappings into Strictly Starlike Domains

- Mathematics
- 2008

AbstractLet Ω be a bounded simply connected domain containing a point w0 and whose boundary is locally connected, $$\mathbb{D} = \left\{ {z:\left| z \right| < 1} \right\}$$ be the open unit disc, and…

Boundary Behavior of Univalent Harmonic Mappings

- Mathematics
- 2014

This chapter is a survey of the boundary behavior properties of univalent harmonic mappings of the unit disk \(\mathbb{D}\) over the last two decades. Particular emphasis has been given to the…

On the Boundary Behaviour of Univalent Harmonic Mappings onto Convex Domains

- Mathematics
- 2008

In this note we discuss the boundary behavior of a univalent harmonic mapping f from the unit disk U “onto” a bounded convex domain Ω in the sense of Hengartner and Schober, whose second dilatation…

Zero-curvature point of minimal graphs

- MathematicsMonatshefte für Mathematik
- 2021

Motivated by a classical result of Finn and Osserman (1964), who proved that the Scherk surface over the square inscribed in the unit disk is extremal for the Gaussian curvature of the point O…

Curvature of minimal graphs

- Mathematics
- 2021

We consider the Gaussian curvature conjecture of a minimal graph S over the unit disk. First of all we reduce the general conjecture to the estimating the Gaussian curvature of some Scherk’s type…

Gaussian curvature conjecture for minimal graphs

- Mathematics
- 2021

In this paper we solve the longstanding Gaussian curvature conjecture of a minimal graph S over the unit disk. This conjecture states the following. For any minimal graph lying above the entire unit…

## References

SHOWING 1-10 OF 22 REFERENCES

Boundary values versus dilatations of harmonic mappings

- Mathematics
- 1997

AbstractThis article is divided into two parts. In the first part, we consider univalent harmonic mappings from the unit diskU onto a Jordan domain Ω whose dilatation functions
$$a = \bar f_{\bar z}…

On harmonic typically real mappings

- Mathematics
- 2003

Abstract In this paper, we prove that a univalent orientation-preserving harmonic mapping f=h+ g defined on the unit disk U with the normalization f(0)=0, h′(0)+ g′(0) >0 , is a typically real…

On the boundary behavior of orientation-preserving harmonic mappings

- Mathematics
- 1986

Nonconstant soluliuns of the partial differential equation where a is analytic in the open unit disk and |a| <1, are orientation-preserving harmonic mappings. We consider the case where ais a finite…

On univalent harmonic mappings and minimal surfaces

- Mathematics
- 2000

If S is the graph of a minimal surface, then when given parametrically by the Weierstrass representation, the first two coordinate functions give a univalent harmonic mapping. In this paper, the…

Harmonic Mappings in the Plane

- Mathematics
- 2004

1. Preliminaries 2. Local properties of harmonic mappings 3. Harmonic mappings onto convex regions 4. Harmonic self-mappings of the disk 5. Harmonic univalent functions 6. Extremal problems 7.…

A constructive method for starlike harmonic mappings

- 1989

SummaryThe idea initiated by Opfer for constructing conformal mappings from the disk onto starlike domains is generalized for univalent harmonic mappings. This is of some interest, since such…

A constructive method for computing the tension parameters in convexity-preserving spline-in-tension interpolation

- Mathematics
- 1988

The idea initiated by Opfer for constructing conformal mappings from the disk onto starlike domains is generalized for univalent harmonic mappings. This is of some interest, since such mappings are…

Bojarskii, Generalized solutions of a system of differential equations of first order and elliptic type with discontinuous coefficients, Mat

- Sb. (N.S.)
- 1957

GENERALIZED SOLUTIONS OF A SYSTEM OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND ELLIPTIC TYPE WITH DISCONTINUOUS COEFFICIENTS

- Mathematics
- 2009

W.Hengartner and O.Hossian, Harmonic typically real mappings, Math

- Proc. Cambridge Philos. Soc.,
- 1996