## Abstract

Let *D* denote the open unit disc and *f* : *D* → be meromorphic and injective in *D*. We further assume that *f* has a simple pole at the point *p* ∈ (0, 1) and an expansion

In particular, we consider functions *f* that map *D* onto a domain whose complement with respect to 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 *C _{o}*(

*p*).

It is proved that for fixed *p* ∈ (0, 1) the domain of variability of the coefficient *a*
_{n}(*f*), *n* ⩾ 2, *f* ∈ *C _{o}*(

*p*), is determined by the inequality

This settles two conjectures published by A. E. Livingston in 1994 and by Ch. Pommerenke and the authors of the present article in 2004.

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