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Harmonic Mappings onto Convex Domains

Published online by Cambridge University Press:  20 November 2018

Yusuf Abu-Muhanna  and
Glenn Schober
Yusuf Abu-Muhanna
Affiliation:
University of Petroleum and Minerals, Dhahran, Saudi Arabia
Glenn Schober
Affiliation:
Indiana University, Bloomington, Indiana
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Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.

In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equation

where a is analytic and |a| < 1, such that f(U) ⊂ D and

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 6 , 01 December 1987 , pp. 1489 - 1530
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Choquet, G., Sur un type de transformation analytique généralisant la représentation. conforme et définie au moyen de fonctions harmoniques, Bull. Sci. Math. 69 (1945), 156–165.Google Scholar
2. Clunie, J. G. and Sheil-Small, T., Harmonie univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I. 9(1984), 3–25.Google Scholar
3. Duren, P. L., Theory of Hp spaces, (Academic Press, 1970).Google Scholar
4. Garnett, J., Bounded analytic functions, (Academic Press, 1981).Google Scholar
5. Hall, R. R., On an inequality of E. Heinz, J. Analyse Math. 42 (1982/83), 185–198.Google Scholar
6. Heinz, E., Über die Lösungen der Minimalflächengleichung, Nachr. Akad. Wiss. Gö;ttingen Math.-Phys. Kl. (1952), 51–56.Google Scholar
7. Hengartner, W. and Schober, G., Harmonic mappings with given dilatation, J. London Math. Soc. 33 (1986), 473–483.Google Scholar
8. Hengartner, W. and Schober, G., Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), 1–31.Google Scholar
9. Hopf, E., On an inequality for minimal surfaces z = z(x, y), J. Rational Mech. Anal. 2 (1953), 519-522 and 801–802.Google Scholar
10. Kneser, H., Losung der Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926), 123–124.Google Scholar
11. Nitsche, J. C. C., Vorlesungen uber Minimalfldchen, (Springer-Verlag, 1975).CrossRefGoogle Scholar
12. Osserman, R., A survey of minimal surfaces, (Dover, 1986).Google Scholar
13. Royster, W. C. and Ziegler, M., Univalent functions convex in one direction, Publ. Math. Debrecen 23 (1976), 339–345.Google Scholar