# Harmonic Mappings in the Plane

@inproceedings{Duren2004HarmonicMI,
title={Harmonic Mappings in the Plane},
author={Peter L. Duren},
year={2004}
}
• P. Duren
• Published 29 March 2004
• Mathematics
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.
583 Citations

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## 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
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
• 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
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
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