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 $f^{(n)}=gh^{(n)}$. 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 $N^+$.