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

Let D denote the open unit disc and let p ∈ (0;1). We consider the family Co(p) of functions 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 is convex. We determine the exact domains of variability of some coefficients an(f) of the Laurent expansion 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. © 2007, Department of Mathematics, Tokyo Institute of Technology. All rights reserved.