Faber–Tietz Functions and Grunsky Coefficients for Mappings into a Torus

Abstract

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.