A proof of the Livingston conjecture

Let D denote the open unit disc and f : D -> (C) over bar be meromorphic and injective in D. We further assume that f has a simple pole at the point p is an element of (0, 1) and an expansion f (z) = z + Sigma(infinity)(n=2) a(n) (f)z(n), vertical bar z vertical bar < p. In particular, we consider functions f that map D onto a domain whose complement with respect to CC is convex. Because of the shape of f (D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p). It is proved that for fixed p is an element of (0, 1) the domain of variability of the coefficient a(n) (f), n >= 2, f is an element of Co(p), is determined by the inequality vertical bar a(n) (f)- 1-p(2n+2)/p(n-1)(1-p(4))vertical bar <= p(2)(1-p(2n-2)/p(n-1)(1-p(4))) This settles two conjectures published by A. E. Livingston in 1994 and by Ch. Pornmerenke and the authors of the present article in 2004.