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 $\pi A$.
Keywords:
concave schlicht functions
Mots-clés :
Taylor coefficients.
F. G. Avkhadiev; K.-J. Wirths. Concave schlicht functions with bounded opening angle at infinity. Lobachevskii journal of mathematics, Tome 17 (2005), pp. 3-10. http://geodesic.mathdoc.fr/item/LJM_2005_17_a0/
@article{LJM_2005_17_a0,
author = {F. G. Avkhadiev and K.-J. Wirths},
title = {Concave schlicht functions with bounded opening angle at infinity},
journal = {Lobachevskii journal of mathematics},
pages = {3--10},
year = {2005},
volume = {17},
language = {en},
url = {http://geodesic.mathdoc.fr/item/LJM_2005_17_a0/}
}
TY - JOUR
AU - F. G. Avkhadiev
AU - K.-J. Wirths
TI - Concave schlicht functions with bounded opening angle at infinity
JO - Lobachevskii journal of mathematics
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[2] F. G. Avkhadiev and K.-J. Wirths, “The conformal radius as a function and its gradient image”, Israel J. of Math., 145 (2005), 349–374 | DOI | MR | Zbl
[3] F. G. Avkhadiev, Ch. Pommerenke and K.-J. Wirths, “Sharp inequalities for the coefficients of concave schlicht functions”, Comment. Math. Helv., 81:4 (2006), 801–807 | MR | Zbl
[4] L. Brickman, D. J. Hallenbeck, T. H. MacGregor and D. R. Wilken, “Convex hulls and extreme points of families of starlike and convex mappings”, Transactions Amer. Math. Soc., 185 (1973), 413–228 | DOI | MR
