Univalent harmonic mappings

ArticleOriginal scientific text

Title

Authors ¹

Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark, Delaware 19716, U.S.A.

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

\bar{S_{H} (U, Ω)}

of

S_{H} (U, Ω)

and determine the extreme points of

\bar{S_{H} (U, Ω)}

.

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