Asymptotic stability for sets of polynomials

Thomas W. Mueller, Jan-Christoph Schlage-Puchta

Thomas W. Mueller, School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, England

Jan-Christoph Schlage-Puchta, Mathematisches Institut, Universitaet Freiburg, Eckerstrasse 1, 79104 Freiburg, Germany



We introduce the concept of asymptotic stability for a set of complex functions analytic around the origin, implicitly contained in an earlier paper of the first mentioned author (``Finite group actions and asymptotic expansion of $e^{P(z)}$", Combinatorica 17 (1997), 523 -- 554). As a consequence of our main result we find that the collection of entire functions $\exp(\mathfrak{P})$ with $\mathfrak{P}$ the set of all real polynomials $P(z)$ satisfying Hayman's condition $[z^n]\exp(P(z))>0\,(n\geq n_0)$ is asymptotically stable. This answers a question raised in loc. cit.

AMSclassification. 30B10.

Keywords. Power series, coefficients, asymptotic expansion.