# Univalent harmonic mappings

Annales Polonici Mathematici (1992)

- Volume: 57, Issue: 1, page 57-70
- ISSN: 0066-2216

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topAlbert E. Livingston. "Univalent harmonic mappings." Annales Polonici Mathematici 57.1 (1992): 57-70. <http://eudml.org/doc/262518>.

@article{AlbertE1992,

abstract = {Let a < 0, Ω = ℂ -(-∞, a] and U = z: |z| < 1. We consider the class $S_H(U,Ω)$ of functions f which are univalent, harmonic and sense preserving with f(U) = Ω and satisfy f(0) = 0, $f_z(0) > 0$ and $f_\{z̅\}(0) = 0$. We describe the closure $\overline\{S_H(U,Ω)\}$ of $S_H(U,Ω)$ and determine the extreme points of $\overline\{S_H(U,Ω)\}$.},

author = {Albert E. Livingston},

journal = {Annales Polonici Mathematici},

keywords = {orientation preserving; extreme points},

language = {eng},

number = {1},

pages = {57-70},

title = {Univalent harmonic mappings},

url = {http://eudml.org/doc/262518},

volume = {57},

year = {1992},

}

TY - JOUR

AU - Albert E. Livingston

TI - Univalent harmonic mappings

JO - Annales Polonici Mathematici

PY - 1992

VL - 57

IS - 1

SP - 57

EP - 70

AB - Let a < 0, Ω = ℂ -(-∞, a] and U = z: |z| < 1. We consider the class $S_H(U,Ω)$ of functions f which are univalent, harmonic and sense preserving with f(U) = Ω and satisfy f(0) = 0, $f_z(0) > 0$ and $f_{z̅}(0) = 0$. We describe the closure $\overline{S_H(U,Ω)}$ of $S_H(U,Ω)$ and determine the extreme points of $\overline{S_H(U,Ω)}$.

LA - eng

KW - orientation preserving; extreme points

UR - http://eudml.org/doc/262518

ER -

## References

top- [1] Y. Abu-Muhanna and G. Schober, Harmonic mappings onto convex domains, Canad. J. Math. 39 (1987), 1489-1530. Zbl0644.30003
- [2] J. A. Cima and A. E. Livingston, Integral smoothness properties of some harmonic mappings, Complex Variables 11 (1989), 95-110. Zbl0724.30011
- [3] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. AI 9 (1984), 3-25. Zbl0506.30007
- [4] D. J. Hallenbeck and T. H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, Monographs and Studies in Math. 22, Pitman, 1984. Zbl0581.30001
- [5] W. Hengartner and G. Schober, Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), 1-31.
- [6] W. Hengartner and G. Schober, Curvature estimates for some minimal surfaces, in: Complex Analysis, Birkhäuser, 1988, 87-100. Zbl0664.30012
- [7] W. Szapiel, Extremal problems for convex sets. Applications to holomorphic functions, Dissertation XXXVII, UMCS Press, Lublin 1986 (in Polish). Zbl0602.30028

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