Factoring derivatives of functions in the Nevanlinna and Smirnov classes

Details Factoring derivatives of functions in the Nevanlinna and Smirnov classes Factoring derivatives of functions in the Nevanlinna and Smirnov classes

2011-09-08

arxiv.org/abs/1109.1753v2

Mathematics

Complex Variables

8 pages; to appear in Annales Academiae Scientiarum Fennicae Mathematica

Scientific paper

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 $N^+$.

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