Annales Academię Scientiarum Fennicę
Mathematica
Volumen 37, 2012, 407-413
ICREA and Universitat de Barcelona, Departament de Matemątica
Aplicada i Anąlisi
Gran Via 585, E-08007 Barcelona, Spain; konstantin.dyakonov 'at' icrea.cat
Abstract. We prove that, given a function f in the Nevanlinna class N and a positive integer n, there exist g \in N and h \in BMOA such that f(n) = gh(n). We may choose g to be zero-free, so it follows that the zero sets for the class N(n) := {f(n) : f \in N} are the same as those for BMOA(n). Furthermore, while the set of all products gh(n) (with g and h as above) is strictly larger than N(n), we show that the gap is not too large, at least when n = 1. Precisely speaking, the class {gh' : g \in N, h \in BMOA} turns out to be the smallest ideal space containing {f' : f \in N}, where "ideal" means invariant under multiplication by H\infty functions. Similar results are established for the Smirnov class N+.
2010 Mathematics Subject Classification: Primary 30D50, 30D55.
Key words: Nevanlinna class, Smirnov class, BMOA, derivatives, factorization, zero sets.
Reference to this article: K.M. Dyakonov: Factoring derivatives of functions in the Nevanlinna and Smirnov classes. Ann. Acad. Sci. Fenn. Math. 37 (2012), 407-413.
doi:10.5186/aasfm.2012.3728
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