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.
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Research supported by Colciencias and Deutsche Forschungsgemeinschaft (DFG).
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Mejía, D., Pommerenke, C. The Analytic Fixed Point Function in the Disk. Comput. Methods Funct. Theory 5, 275–299 (2006). https://doi.org/10.1007/BF03321099
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DOI: https://doi.org/10.1007/BF03321099
Keywords
- Fixed point function
- byperbolically convex
- coefficients
- asymptotic behaviour
- probability generating function
- large deviations
- branching process