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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.40 no.1 Bogotá Jan./June 2006
Diego Mejía*
Escuela de Matemáticas
Universidad Nacional de Colombia. A.A. 3840. Medellin, Colombia
e-mail: dmejia@unal.edu.co
* Supported by COLCIENCIAS.
Christian Pommerenkey+
Institut für Mathematik MA 8-2. Technische Universität. D-10623. Berlin, Germany
e-mail: pommeren@math.tu-berlin.de
+ Supported by Deutsche Forschungsgemeinschaft (DFG).
ABSTRACT. Let φ be analytic in the unit disk D and let φ(D) ⊂ D, φ(0) 6= 0. Then w = z/φ(z) has an analytic inverse z = f(w) for w ∈ D, the fixed point function. This paper studies the case that φ(1) = φ´(1) = 1 with a growth condition for φ´´(x) and determines the asymptotic behaviour of various combinations of the coefficients of φ connected with f. The results can be interpreted in various contexts of probability theory.
Keywords and phrases. Fixed point function, coefficients, Bürmann-Lagrange, asymptotics, equilibrium, first return, branching process.
2000 Mathematics Subject Classification. Primary: 30B10. Secondary: 60F99, 60J80.
RESUMEN. Sea φ analítica en el disco unitario D y φ(D) ⊂ D; φ(0) 6= 0. Entonces w = z/φ(z) tiene una inversa analítica z = f(w) para w ∈ D, la función de punto fijo. Este artículo estudia el caso en que φ(1) = φ´(1) = 1 con una condición de crecimiento para φ´´(x) y determina el comportamiento asintótico de varias combinaciones de los coeficientes de φ conectados con f. Los resultados se pueden interpretar en varios contextos de la teoría de la probabilidad.
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(Recibido en febrero de 2006. Aceptado en abril de 2006)