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## A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

### Tom 83 / 2004

Annales Polonici Mathematici 83 (2004), 87-93 MSC: 30C50, 30C45. DOI: 10.4064/ap83-1-10

#### Streszczenie

Let $D$ denote the open unit disc and $f:D\to \overline{\mathbb C}$ be meromorphic and injective in $D$. We further assume that $f$ has a simple pole at the point $p\in (0,1)$ and an expansion $f(z)=z+\sum_{n=2}^{\infty}a_n(f)z^n, \quad |z|< p.$ In particular, we consider $f$ that map $D$ onto a domain whose complement with respect to $\overline{\mathbb C}$ is convex. Because of the shape of $f(D)$ these functions will be called concave univalent functions with pole $p$ and the family of these functions is denoted by ${\rm Co}(p)$. It is proved that for $p\in (0,1)$ the domain of variability of the coefficient $a_n(f)$, $f\in {\rm Co}(p),$ for $n\in \{2,3,4,5\}$ is determined by the inequality $\biggl|a_n(f) - \frac{1 - p^{2n+2}}{p^{n-1}(1-p^4)}\biggr| \leq\frac{p^2(1 - p^{2n-2})}{p^{n-1}(1-p^4)}.$ In the said cases, this settles a conjecture from [1]. The above inequality was proved for $n=2$ in [6] and [2] by different methods and for $n=3$ in [1]. A consequence of this inequality is the so called Livingston conjecture (see [4]) ${\rm Re}(a_n(f))\geq \frac{1+p^{2n}}{p^{n-1}(1+p^2)}.$

#### Autorzy

• Karl-Joachim WirthsInstitut für Analysis
TU Braunschweig
38106 Braunschweig, Germany
e-mail

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