Factoring derivatives of functions in the Nevanlinna and Smirnov classes

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 $f^{(n)}=g{h}^{(n)}$. 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 $BMO{A}^{(n)}$. Furthermore, while the set of all products $g{h}^{(n)}$ (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 $\{g{h}^{\prime }:g\in N,h\in BMOA\}$ turns out to be the smallest ideal space containing $\{f^{\prime }:f\in N\}$, where "ideal" means invariant under multiplication by $H^{\infty }$ functions. Similar results are established for the Smirnov class $N^{+}$.

arXiv comment8 pages; to appear in Annales Academiae Scientiarum Fennicae Mathematica

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