Factoring derivatives of functions in the Nevanlinna and Smirnov classes

الأصل · EN

We prove that, given a function f in the Nevanlinna class N and a positive integer n, there exist g∈ N and h∈ 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∈ 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∈ N, h∈ BMOA} turns out to be the smallest ideal space containing {f': f∈ N}, where "ideal" means invariant under multiplication by H∞ functions. Similar results are established for the Smirnov class N+.