A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions
Volume 83 / 2004
Abstract
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]) \[ {\rm Re}(a_n(f))\geq \frac{1+p^{2n}}{p^{n-1}(1+p^2)}. \]
