Geodesic
Concave schlicht functions with bounded opening angle at infinity
Lobachevskii journal of mathematics, Tome 17 (2005), pp. 3-10.

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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. 
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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/

[1] F. G. Avkhadiev and K.-J. Wirths, “Convex holes produce lower bounds for coefficients”, Compl. Var., 47 (2002), 553–563 | DOI | MR | Zbl

[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

[5] K.-J. Wirths, “Julia's Lemma and concave schlicht functions”, Quaestiones Mathematicae, 28 (2005), 1–9 | MR