Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Next Article Computational Methods and Function Theory 5 (2005), No. 2, 275--299 Copyright Heldermann Verlag 2005 The Analytic Fixed Point Function in the Disk Diego Mejía dmejia@unalmed.edu.co , Universidad Nacional, Departamento de Matemáticas, A.A. 3840, Medellin, Colombia. Christian Pommerenke pommeren@math.tu-berlin.de , Technische Universität, Institut für Mathematik, MA 8-2, 10623 Berlin, Germany. [Abstract-pdf] [Abstract-ps] Let $\varphi$ be analytic in the unit disk $\mathbb{D}$ and let $\varphi(\mathbb{D})\subset\mathbb{D}$, $\varphi(0)\neq0$. Then $w=z/\varphi(z)$ has an analytic inverse $z=f(w)$, $w\in\mathbb{D}$, the fixed point function. Here $f(\mathbb{D})$ is a starlike domain and various results suggest that $f(\mathbb{D})$ might even be hyperbolically convex. We study the derivative and the coefficients of $f$, in particular their asymptotic behaviour. In the case that $\varphi$ is the generating function of a random variable, several functions related to $f$ have probabilistic interpretations. Keywords: Fixed point function, byperbolically convex, coefficients, asymptotic behaviour, probability generating function, large deviations, branching process. MSC 2000: 30D50, 30D05, 60F10. [FullText-pdf (370 K)] [FullText-ps (612 K)]