Let a < 0, Ω = ℂ -(-∞, a] and U = z: |z| < 1. We consider the class of functions f which are univalent, harmonic and sense preserving with f(U) = Ω and satisfy f(0) = 0, and . We describe the closure of and determine the extreme points of .
@article{bwmeta1.element.bwnjournal-article-apmv57z1p57bwm,
author = {Albert E. Livingston},
title = {Univalent harmonic mappings},
journal = {Annales Polonici Mathematici},
volume = {57},
year = {1992},
pages = {57-70},
zbl = {0754.30013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv57z1p57bwm}
}
Albert E. Livingston. Univalent harmonic mappings. Annales Polonici Mathematici, Tome 57 (1992) pp. 57-70. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv57z1p57bwm/
[000] [1] Y. Abu-Muhanna and G. Schober, Harmonic mappings onto convex domains, Canad. J. Math. 39 (1987), 1489-1530. | Zbl 0644.30003
[001] [2] J. A. Cima and A. E. Livingston, Integral smoothness properties of some harmonic mappings, Complex Variables 11 (1989), 95-110. | Zbl 0724.30011
[002] [3] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. AI 9 (1984), 3-25. | Zbl 0506.30007
[003] [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. | Zbl 0581.30001
[004] [5] W. Hengartner and G. Schober, Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), 1-31.
[005] [6] W. Hengartner and G. Schober, Curvature estimates for some minimal surfaces, in: Complex Analysis, Birkhäuser, 1988, 87-100. | Zbl 0664.30012
[006] [7] W. Szapiel, Extremal problems for convex sets. Applications to holomorphic functions, Dissertation XXXVII, UMCS Press, Lublin 1986 (in Polish). | Zbl 0602.30028