Concave schlicht functions with bounded opening angle at infinity
(Lobachevskii Journal of Mathematics, Volume XVII)
Let D denote the open unit disc. In this article we consider functions f(z)=z + Σn=2∞an(f)zn that map D conformally onto a domain whose complement with respect to C is convex and that satisfy the normalization f(1)=∞. Furthermore, we impose on these functions the condition that the opening angle of f(D) at infinity is less than or equal to π A, A∈(1,2]. We will denote these families of functions by CO(A). Generalizing the results of [AW1], [APW2], and [W1], where the case A=2 has been considered, we get representation formulas for the functions in CO(A). They enable us to derive the exact domains of variability of a2(f) and a3(f), f∈CO(A). It turns out that the boundaries of these domains in both cases are described by the coefficients of the conformal maps of D onto angular domains with opening angle π A.