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

Karl-Joachim Wirths¹

¹ Institut für Analysis TU Braunschweig 38106 Braunschweig, Germany

Annales Polonici Mathematici, Tome 83 (2004) no. 1, pp. 87-93

Résumé

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]) \[ {\rm Re}(a_n(f))\geq \frac{1+p^{2n}}{p^{n-1}(1+p^2)}. \]

Détail

DOI : 10.4064/ap83-1-10

Keywords: denote unit disc overline mathbb meromorphic injective further assume has simple pole point expansion sum infty z quad particular consider map domain whose complement respect overline mathbb convex because shape these functions called concave univalent functions pole family these functions denoted proved domain variability coefficient determined inequality biggl frac n p biggr leq frac n n p said cases settles conjecture above inequality proved different methods consequence inequality called livingston conjecture see geq frac n