# 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 + Σ*_{n=2}^{∞}a_{n}(f)z^{n} 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 *a*_{2}(f) and *a*_{3}(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*.