# Harmonic Mappings in the Plane

@inproceedings{Duren2004HarmonicMI, title={Harmonic Mappings in the Plane}, author={Peter L. Duren}, year={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. Mapping problems 8. Additional topics 9. Minimal surfaces 10. Curvature of minimal surfaces Appendix References.

## Figures from this paper

## 583 Citations

Convex harmonic mappings are not necessarily in h 1/2

- Mathematics
- 2014

We construct convex harmonic mappings in the unit disk that do not belong to the harmonic Hardy space h(1/2). This provides a negative answer to a question raised by P. Duren.

Curvature Properties of Planar Harmonic Mappings

- Mathematics
- 2004

A geometric interpretation of the Schwarzian of a harmonic mapping is given in terms of geodesic curvature on the associated minimal surface, generalizing a classical formula for analytic functions.…

Harmonic Mappings Related to Janowski Convex Functions of Complex Order b

- Mathematics
- 2013

Let SH be the class of all sense-preserving harmonic mappings in the open unit disc D = {z ∈ C||z| < 1}. In the present paper the authors investigate the properties of the class of harmonic mappings…

Recent Results on Harmonic and p-harmonic Mappings

- Mathematics
- 2011

The class of conformal mappings is a natural object in the study of theory of analytic functions. Likewise univalent harmonic mappings (both in the planar and in the higher dimensional case) is…

ON PLANAR HARMONIC LIPSCHITZ AND PLANAR HARMONIC HARDY CLASSES

- Mathematics
- 2011

In this paper, we investigate the properties of two classes of planar harmonic map- pings. First, we discuss the equivalent norms on Lipschitz-type spaces of harmonic K-quasiregular mappings and then…

Inclusion properties of planar harmonic mappings associated with the Wright function

- MathematicsComplex Variables and Elliptic Equations
- 2020

ABSTRACT The prime purpose of this article aims to establish connections between various sub-classes of harmonic univalent mappings by applying a convolution operator associated with the well-known…

INTEGRAL MEANS AND COEFFICIENT ESTIMATES ON PLANAR HARMONIC MAPPINGS

- Mathematics
- 2012

In this paper, we investigate some properties of planar harmonic mappings in Hardy spaces. First, we discuss the integral means of harmonic mappings and those of their derivatives, and as a…

QUASICONFORMAL EXTENSION OF HARMONIC MAPPINGS IN THE PLANE

- Mathematics
- 2013

Let f be a sense-preserving harmonic mapping in the unit disk. We give a sufficient condition in terms of the pre-Schwarzian derivative of f to ensure that it can be extended to a quasiconformal map…

Bounded harmonic mappings related to starlike functions

- Mathematics
- 2014

Let f = h(z)+g(z)¯ be a sense-preserving harmonic mapping in the open unit disc D = {z||z| 12′ then f is called bounded harmonic mapping. The main purpose of this paper is to give some properties of…

On locally invertible harmonic mappings of plane domains

- Mathematics
- 2017

We prove several global univalence theorems for locally invertible harmonic mappings with certain prescribed boundary behavior in simply and multiply connected domains. In particular, we consider…

## References

SHOWING 1-10 OF 117 REFERENCES

A variational method for harmonic mappings onto convex regions

- Mathematics
- 1987

Harmonic mappings arise in several areas of analysis and geometry. Our purpose is to present a general variational method for treating extremal problems over families of univalent harmonic functions…

Estimates of integral means of harmonic mappings

- Mathematics
- 2000

In this paper we prove that each univalent harmonic mapping of the unit disc onto a given horizontal strip is in h 1 and we find a uniform upper bound for their h 1-norm. We also give some estimates…

Harmonic typically real mappings

- Mathematics
- 1996

We give an example of a univalent orientation-preserving harmonic mapping f = h + g¯ defined on the unit disc U which is real on the real axis, satisfies and is not typically real. Furthermore, we…

Planar harmonic mappings and curvature estimates

- Mathematics
- 1995

Let $\Sigma$ be the class of all complex-valued, harmonic, orientation-preserving, univalent mappings defined on $\Delta = {z : z > 1}$ that map $\infty$ to $\infty$.

Univalent Harmonic Ring Mappings Vanishing on the Interior Boundary

- MathematicsCanadian Journal of Mathematics
- 1992

Abstract We give a characterization of univalent positively oriented harmonic mappings ƒ defined on an exterior neighbourhood of the closed unit disk { z: | z| ≤1} such that .

On the dilatation of univalent planar harmonic mappings

- Mathematics
- 1998

It is shown that if f is a univalent harmonic mapping of the unit disk onto a domain having a smooth boundary arc which is convex with respect to the domain, and if the dilatation has modulus 1 on…

Local decomposition of harmonic mappings

- Mathematics
- 1987

Under certain conditions a two-dimensional harmonic mapping h can be represented nontrivially as h=g-f where f, g are both harmonic. On the basis of the current work, in conjunction with previous…

Integral smoothness properties of some harmonic mappings

- Mathematics
- 1989

Let be a univalent mapping of the unit disk (suitably normalized) onto a strip. We prove that the analytic completion of u, is the Cauihy trdnsiorm of a mcasure on the unii circle. It is also proven…

Boundary correspondence and dilatation of harmonic mappings

- Mathematics
- 1997

If a sense-preserving harmonic mapping has dilatation of unit modulus on some boundary are, then it maps that are onto a concave are unless it is piecewise constant A theorem to this effect is proved…

Construction of close-to-convex harmonic polynomials

- Mathematics
- 2001

A family of sense–preserving complex valued harmonic polynomials that are close-to-convex on the open unit disk is introduced. By taking limits as the degree of the polynomials tends to infinity, a…