抄録
Let D denote the open unit disc and let p ∈ (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) = $¥sum_{n=-1}^{¥infty}$ an (f)(z − p)n, |z − p| < 1 − p,
for f ∈ 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.
