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Behavior of Coefficients of Inverses of α-Spiral Functions

Published online by Cambridge University Press:  20 November 2018

Richard J. Libera  and
Eligiusz J. Złotkiewicz
Richard J. Libera
Affiliation:
University of Delaware, Newark, Delaware
Eligiusz J. Złotkiewicz
Affiliation:
Uniwersytet Marii Curie-Sklodowskiej, Lublin, Polska
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If f(z) is univalent (regular and one-to-one) in the open unit disk Δ, Δ = {z ∊ C:│z│ < 1}, and has a Maclaurin series expansion of the form

(1.1)

then, as de Branges has shown, │ak│ = k, for k = 2, 3, … and the Koebe function.

(1.1)

serves to show that these bounds are the best ones possible (see [3]). The functions defined above are generally said to constitute the class .

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 6 , 01 December 1986 , pp. 1329 - 1337
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Campschroer, J. T. P., Coefficients of the inverse of a convex function, Report 8227, Dept. of Math., Catholic University, Nijmegen, The Netherlands (1982).Google Scholar
2. Dwight, H. B., Tables of integrals and other mathematical data (New York, 1961).Google Scholar
3. de Branges, L., A proof of the Bieberbach conjecture, Acta Math 154 (1985), 137–152.Google Scholar
4. Grenander, U. and Szegö, G., Toeplitz forms and their applications (Univ. of California Press, Berkeley and Los Angeles, 1958).CrossRefGoogle Scholar
5. Kirwan, W. E. and Schober, G., Inverse coefficients for functions of bounded boundary rotation, J. Analyse Math. 36 (1979), 167–178.Google Scholar
6. Krzyz, J. G., Libera, R. J. and Zlotkiewicz, E. J., Coefficients of inverses of regular starlike functions, Ann. Univ. Mariae Curie-Sktodowska, Sect. A 33 (1979), 103–110.Google Scholar
7. Libera, R. J., Univalent α-spiral functions, Can. J. Math. 19 (1967), 449–456.Google Scholar
8. Libera, R. J. and Zlotkiewicz, E. J., Early coefficients of the inverse of a regular convex function, Proc. A.M.S. 85 (1982), 225–230.Google Scholar
9. Libera, R. J. and Zlotkiewicz, E. J., Coefficient bounds for the inverse of a function with derivative in , Proc. A.M.S. 87 (1983), 251–257.Google Scholar
10. Loewner, C., Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I, Math. Ann. 89 (1923), 103–121.Google Scholar
11. Royden, H. L., Real analysis, 2nd ed. (The Macmillan Company, New York, 1968).Google Scholar
12. Schober, G., Coefficient estimates for inverses of schlicht functions, Aspects of contemporary complex analysis (Academic Press, New York, 1980), 503–513.Google Scholar
13. Špaček, L., Prǐspěvek k teorii funkci prostych, Cǎsopis Pěst. Mat a Fys. 62 (1933), 12–19.Google Scholar