ON SOME INTEGRAL REPRESENTATIONAL FORMULAS FOR SCHWARZIAN COEFFICIENTS WITH AN APPLICATION TO NUMBER THEORY
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Abstract
We develop integral representational formulas for all Schwarzian coefficients of single-slit mappings by utilizing Lowner's Parametric Method. As an application, we evaluate a complicated number theoretic sum.
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References
P. L. Duren, "Univalent Functions", Springer-Verlag, Heidelberg and New York, 1983.
G. M. Goluzin, "Geometric Theory of Functions of a Complex Variable", English Transl., Amer. Math Soc., Providence, R. I., 1969.
R. Harmelin, "Generalized Grunsky coefficient.sand inequalities", Israel J. of Math., Vol. 57, No. 3. (1987), 317-364.
E. Hille, "Analytic Function Theory", II, Chelsea, New York, 1962.
J. A. Hummel, "The Grunsky coefficients of a schlicht function", Proc. Amer. Math. Soc., 15 {1964), 142-150.
J. A. Jenkins, "On certain coefficients of univalent functions in Analytic Functions" Princeton Univ. Press, Princeton, N.J ., 1960.
W. Kraus, "Uger den Zusarmmenhang einiger Charakteristiken eines einfach zusammen- hängenden Bereiches mit der Kreisabbildung", Mitt. Math. Sem. Giessen, 21 (1932), 1-28.
Ch. Pommerenke, "Univalent Functions", Vandenhoeck and Ruprecht Gottingen, 1975.
M. Schiffer, "Sur un probleme d'extremum de la representation conforme", Bull. Soc. Math. France, 66 (1938), 48-55.
G. Schober, "Univalent Functions-Selected Topics", Lecture Notes in Math. No. 478, Springer-Verlag, 1975.
I. Schur, "On Faber Polynomials", Amer. J. Math., 67 (1945), 33-41.
P. Todorov, "Explicit Formulas for the coefficients of Faber polynomials with respect to univalent functions of the class I:",Proc. Amer. Math. Soc., Vol. 82, Number 3, (1981), 431-438.
C. Fitzgerald and Ch. Pommerenke, "The de Branges theorem on univalent functions", Trans. Amer. Math. Soc., 290 (1985), 683-690.
S. Zemyan, "On the Schwarzian Coefficients of Univalent Functions", Bull. Austral. Math. Soc., Vol. 46 (1992), 389-198.