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Faber-Tietz functions and Grunsky coefficients for mappings into a torus. (English) Zbl 1326.30021

Summary: Tietz defined a generalization of Faber polynomials for conformal maps into a compact Riemann surface, which we will call the Faber-Tietz functions. We give a generating function for the Faber-Tietz functions of any torus in terms of the Weierstrass zeta function. We also give explicit expressions for the first few Faber-Tietz functions on the torus in terms of the Weierstrass \(\wp\)-function and Eisenstein series. We also prove a sharp generalization of the Grunsky inequalities for conformal maps into compact Riemann surfaces. We also derive explicit expressions for some of the Grunsky coefficients of a conformal map into a torus.

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable
30C35 General theory of conformal mappings
30F30 Differentials on Riemann surfaces
30B99 Series expansions of functions of one complex variable
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