Concave schlicht functions with bounded opening angle at infinity

F. G. Avkhadiev ; K.-J. Wirths

Lobachevskii journal of mathematics, Tome 17 (2005), pp. 3-10

Résumé

Let $D$ denote the open unit disc. In this article we consider functions $f(z)=z+\sum_{n=2}^\infty a_n(f)z^n$ that map $D$ conformally onto a domain whose complement with respect to $\mathbb C$ is convex and that satisfy the normalization $f(1)=\infty$. Furthermore, we impose on these functions the condition that the opening angle of $f(D)$ at infinity is less than or equal to $\pi A$, $A\in(1,2]$. We will denote these families of functions by $CO(A)$. Generalizing the results of [1], [3], and [5], 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 $a_3(f)$, $f\in CO(A)$. 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 $\pi A$.

Détail

Keywords: concave schlicht functions
Mots-clés : Taylor coefficients.