Domains of variability of Laurent coefficients and the convex hull for the family of concave univalent functions

Abstract

Let D denote the open unit disc and let p $\in$ (0,1). We consider the family Co(p) of functions f : D → $\overline{{\mathbf C}}$ that satisfy the following conditions:

(i) f is meromorphic in D and has a simple pole at the point p.

(ii) f(0) = f′(0) – 1 = 0.

(iii) f maps D conformally onto a set whose complement with respect to $\overline{{\mathbf C}}$ is convex.

We determine the exact domains of variability of some coefficients a_n (f) of the Laurent expansion

$$f(z)=\sum_{n=-1}^{\infty} a_n(f)(z-p)^n, \quad |z – p|<1 – p,$$

for f $\in$ Co(p) and certain values of p. Knowledge on these Laurent coefficients is used to disprove a conjecture of the third author on the closed convex hull of Co(p) for certain values of p.