Univalent harmonic mappings

Albert E. Livingston

Annales Polonici Mathematici (1992)

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

Abstract

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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 z ̅ ( 0 ) = 0 . We describe the closure S H ( U , Ω ) ¯ of S H ( U , Ω ) and determine the extreme points of S H ( U , Ω ) ¯ .

How to cite

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Albert 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

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  1. [1] Y. Abu-Muhanna and G. Schober, Harmonic mappings onto convex domains, Canad. J. Math. 39 (1987), 1489-1530. Zbl0644.30003
  2. [2] J. A. Cima and A. E. Livingston, Integral smoothness properties of some harmonic mappings, Complex Variables 11 (1989), 95-110. Zbl0724.30011
  3. [3] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. AI 9 (1984), 3-25. Zbl0506.30007
  4. [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. [5] W. Hengartner and G. Schober, Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), 1-31. 
  6. [6] W. Hengartner and G. Schober, Curvature estimates for some minimal surfaces, in: Complex Analysis, Birkhäuser, 1988, 87-100. Zbl0664.30012
  7. [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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