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 $[z^n]\exp (P(z))>0\,(n\ge n_0)$ is asymptotically stable. This answers a question raised in loc. cit.
@article{107945,
author = {Thomas W. M\"uller and Jan-Christoph Schlage-Puchta},
title = {Asymptotic stability for sets of polynomials},
journal = {Archivum Mathematicum},
volume = {041},
year = {2005},
pages = {151-155},
zbl = {1109.30001},
mrnumber = {2164664},
language = {en},
url = {http://dml.mathdoc.fr/item/107945}
}
Müller, Thomas W.; Schlage-Puchta, Jan-Christoph. Asymptotic stability for sets of polynomials. Archivum Mathematicum, Tome 041 (2005) pp. 151-155. http://gdmltest.u-ga.fr/item/107945/
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