# The Analytic Fixed Point Function in the Disk

The Analytic Fixed Point Function in the Disk Let φ be analytic in the unit disk D and let φ(D) ⊂ D, φ(0) ≠ 0. Then ω = z/φ(z) has an analytic inverse z = f(ω), ω ∈ D, the fixed point function. Here f(D) is a starlike domain and various results suggest that f(D) might even be hyperbolically convex. We study the derivative and the coefficients of f, in particular their asymptotic behaviour. In the case that φ is the generating function of a random variable, several functions related to f have probabilistic interpretations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# The Analytic Fixed Point Function in the Disk

, Volume 5 (2) – Mar 7, 2013
25 pages      /lp/springer-journals/the-analytic-fixed-point-function-in-the-disk-cEV8KcCe04
Publisher
Springer Journals
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321099
Publisher site
See Article on Publisher Site

### Abstract

Let φ be analytic in the unit disk D and let φ(D) ⊂ D, φ(0) ≠ 0. Then ω = z/φ(z) has an analytic inverse z = f(ω), ω ∈ D, the fixed point function. Here f(D) is a starlike domain and various results suggest that f(D) might even be hyperbolically convex. We study the derivative and the coefficients of f, in particular their asymptotic behaviour. In the case that φ is the generating function of a random variable, several functions related to f have probabilistic interpretations.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 7, 2013

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