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On the Schwarzian coefficients of univalent functions

Published online by Cambridge University Press:  17 April 2009

Stephen M. Zemyan
Stephen M. Zemyan
Affiliation:
Department of Mathematics, Penn State University, Mont Alto PA 17237-9799, United States of America
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Abstract

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For f ∈ S, we study the Schwarzian coefficients sn defined by {f, z} = Σ snzn. Sharp bounds on s0, s1 and s2 are given, together with an order of growth estimate as n → ∞. We use the Grunsky Inequalities to estimate combinations of coefficients.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 3 , December 1992 , pp. 391 - 400
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Duren, P.L., Univalent functions (Springer-Verlag, Berlin, Heidelberg and New York, 1983).Google Scholar
[2]Goluzin, G.M., Geometric theory of functions of a complex variable, English Translation (Amer. Math. Soc., Providence, R.I., 1969).CrossRefGoogle Scholar
[3]Harmelin, R., ‘Generalized Grunsky coefficients and inequalities’, Israel J. Math. 57 (1987), 347–364.CrossRefGoogle Scholar
[4]Hille, E., Analytic Function Theory (Chelsea, New York, 1962).Google Scholar
[5]Hummel, J.A., ‘The Grunsky coefficients of a schlicht function’, Proc. Amer. Math. Soc. 15 (1964), 142–150.CrossRefGoogle Scholar
[6]Jenkins, J.A., ‘On certain coefficients of univalent functions’, in Analytic functions (Princeton University Press, Princeton, N.J., 1960).Google Scholar
[7]Kraus, W., ‘Über den Zusammenhang einiger Charakteristiken eines einfach zusammenhängenden Bereiches mit der Kreisabbildung’, Mitt. Math. Sem. Giessen 21 (1932), 1–28.Google Scholar
[8]Pommerenke, Ch., Univalent Functions (Vandenhoeck and Ruprecht, Göttingen, 1975).Google Scholar
[9]Schiffer, M., ‘Sur un problème d'extremum de la représentation conforme’, Bull. Soc. Math. France 66 (1938), 48–55.CrossRefGoogle Scholar
[10]Schober, G., Univalent Functions-Selected Topics: Lecture Notes in Math. 478 (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[11]Schur, I., ‘On Faber polynomials’, Amer. J. Math. 67 (1945), 33–41.CrossRefGoogle Scholar
[12]Todorov, P., ‘Explicit Formulas for the coefficients of Faber polynomials with respect to univalent functions of the class Σ’, Proc. Amer. Math. Soc. 82 (1981), 431–438.Google Scholar