Factoring derivatives of functions in the Nevanlinna and Smirnov classes

TL;DR

This paper demonstrates that derivatives of functions in the Nevanlinna and Smirnov classes can be factored into products involving functions from these classes and BMOA, revealing structural similarities and differences in their zero sets and ideal spaces.

Contribution

It introduces a novel factorization of derivatives in the Nevanlinna and Smirnov classes, linking their zero sets and ideal spaces with BMOA functions.

Findings

Zero sets for derivatives in $N$ and $BMOA$ coincide.

The class of products $gh'$ is the minimal ideal space containing derivatives of $N$ functions.

Similar factorization results hold for the Smirnov class $N^+$.

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 $N^+$.