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Asymptotic stability for sets of polynomials

Thomas W. Müller, Jan-Christoph Schlage-Puchta (2005)

Archivum Mathematicum

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 ( 𝔓 ) with 𝔓 the set of all real polynomials P ( z ) satisfying Hayman’s condition [ z n ] exp ( P ( z ) ) > 0 ( n n 0 ) is asymptotically stable. This answers a question raised in loc. cit.

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