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Let $\ID$ denote the open unit disc and let $p\in (0,1)$. We consider thefamily $Co(p)$ of functions $f:\ID\to \overline{\IC}$ that satisfy thefollowing conditions: \bee \item[(i)] $f$ is meromorphic in $\ID$ and has asimple pole at the point $p$. \item[(ii)] $f(0)=f'(0)-1=0$. \item[(iii)] $f$maps $\ID$ conformally onto a set whose complement with respect to$\overline{\IC}$ is convex. \eee We determine the exact domains of variabilityof 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 todisprove a conjecture of the third author on the closed convex hull of $Co(p)$for certain values of $p$.

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