Explicit formulas for the coefficients of Faber polynomials with respect to univalent functions of the class Σ

Todorov, Pavel G.

doi:10.1090/S0002-9939-1981-0612735-4

Explicit formulas for the coefficients of Faber polynomials with respect to univalent functions of the class $\Sigma$
HTML articles powered by AMS MathViewer

by Pavel G. Todorov PDF

Proc. Amer. Math. Soc. 82 (1981), 431-438 Request permission

Abstract:

In this note, we obtain three natural explicit formulas for the coefficients of the Faber polynomials with respect to the univalent functions of the class $\sum$. As an application, we obtain two new explicit formulas for the Grunsky coefficients of functions in $\sum$; these formulas are simpler than those due to Schur [2] and Hummel [4]. The method used in this paper is different from the one that we used to obtain explicit expressions for the Grunsky coefficients for functions of the class $S$ [6].

References

Helmut Grunsky, Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen, Math. Z. 45 (1939), no. 1, 29–61 (German). MR 1545803, DOI 10.1007/BF01580272
Issai Schur, On Faber polynomials, Amer. J. Math. 67 (1945), 33–41. MR 11740, DOI 10.2307/2371913
Menahem Schiffer, Faber polynomials in the theory of univalent functions, Bull. Amer. Math. Soc. 54 (1948), 503–517. MR 25564, DOI 10.1090/S0002-9904-1948-09027-9
J. A. Hummel, The Grunsky coefficients of a schlicht function, Proc. Amer. Math. Soc. 15 (1964), 142–150. MR 158060, DOI 10.1090/S0002-9939-1964-0158060-X
Pavel G. Todorov, New explicit formulas for the coefficients of $p$-symmetric functions, Proc. Amer. Math. Soc. 77 (1979), no. 1, 81–86. MR 539635, DOI 10.1090/S0002-9939-1979-0539635-3

New explicit formulas for the Grunsky coefficients of univalent functions