Vector Fields on the Space of Functions Univalent Inside the Unit Disk via Faber Polynomials

On the asymptotic behavior of Faber polynomials for domains with piecewise analytic boundary

For a function g(w) analytic and univalent in {w:1<|w|<\infty} with a simple pole at \infty and a continuous extension to {w:|w|\geq 1}, we consider the Faber polynomials F_n(z), n=0,1,2,..., associated to g(w) via their generating function g'(w)/(g(w)-z)=\sum_{n=0}^\infty F_n(z)w^{-(n+1)}. Assuming that g(w) maps the unit circle T onto a piecewise analytic curve L whose exterior domain has no outward-pointing cusps, and under an additional assumption concerning the "Lehman expansion" of g(w) about those points of T mapped onto corners of L, we obtain asymptotic formulas for F_n(z) that yield …