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 an (f) of the Laurent expansion ¶ $f(z)=\sum_{n=-1}^{\infty} a_n(f)(z-p)^n,$ |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.